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Institute of Numerical Mathematics
Russian Academy of Sciences
II INTERNATIONAL CONFERENCE
ON MATRIX METHODS
AND OPERATOR EQUATIONS
Web-site: bach.inm.ras.ruE-mail: "E. E. Tyrtyshnikov" <[email protected]>
PRINCIPAL ORGANIZERS:
Institute of Numerical Mathematics, Russian Academy of SciencesLomonosov Moscow State University
CO-ORGANIZERS & SPONSORS:
Russian Foundation for Basic ResearchInternational Foundation for Technology and Investments (IFTI)Neurok TechsoftUniversity of Insubria (Como, Italy)
ORGANIZING COMMITTEE:
Dario Bini University of Pisa, ItalyGene Golub Stanford University, USAAlexander Guterman Moscow State UniversityYuri Nesterenko Moscow State UniversityVadim Olshevsky University of Connecticut, USAStefano Serra-Capizzano University of Insubria, ItalyGilbert Strang M.I.T., USAEugene Tyrtyshnikov INM RAS; Moscow State University
APPLICATION SECTION COMMITTEE:
O. Diyankov (Neurok), S. Karabashev (IFTI), M. Olshansky (MSU), Yu. Vassilevski (INM RAS)
ABSTRACTS
Moscow, July 23–27, 2007
On the spectra of the Laplacian matrices of a certain class
Rafig AgaevInstitute of Control Sciences of the Russian Academy of Science, Moscow
A Laplacian matrix L = (ℓij) ∈ Rn×n has nonpositive off-diagonal entries and zero row sums.Let Lα = (lij) be a Laplacian matrix defined as
1 2 . . . i i+ 1 . . . n− 1 n
Lα =
2 −1 −1−1 2
. . .1 + α −α−1 2
. . .2 −1−1 1
12. . .ii+ 1. . .n− 1n
In this paper the spectra of the matrices L1 and L0 are studied by means of the Chebyshevpolynomials Pn(x) of the second kind. More particularly, we consider the polynomials Zn(x), n ∈ N,that satisfy the recurrence relation Zn(x) = (x − 2)Zn−1(x) − Zn−2(x) with the initial conditionsZ0(x) ≡ 1 and Z1(x) ≡ x− 1, in which case Zn(x) and Pn(x) are related by Zn(x) = P2n(
√x).
Theorem 1. The characteristic polynomial of L1 is ∆L1(λ) = Zn(λ)−(−1)n and the eigenvaluesof L1 are 4 cos2( πk
2n+1−(−1)k+n ), k = 1, . . . , n.
Theorem 2. The characteristic polynomial of L0 is ∆L0(λ) = Zi(λ)Zn−i(λ) − (−1)n. Foran even n, all eigenvalues of L0 are real if and only if i = n
2 , in which case they are 4 cos2(πkn ),
4 cos2( πkn+2), k = 1, . . . , n
2 . For an odd n, all eigenvalues of L0 are real if and only if i = n−12 or
i = n+12 ; in either case they are 0 and 4 cos2( πk
n+1 ), k = 1, . . . , n−12 , of multiplicity two.
In this work we also show that the theorems 1 and 2 can be applied to calculation of the numberof spanning trees of digraphs of a certain class. A similar result for undirected graphs is due toBoesch and Prodinger [1] and Zhang, Yong and Golin [2].
References1. Boesch F.T., Prodinger H. Spanning tree formulas and Chebyshev polynomials, Graphs
and Combinatorics. 1986. V. 2. P. 191–200.2. Zhang Y., Yong X., Golin M. Chebyshev polynomials and spanning tree formulas for
circulant and related graphs Discrete Mathematics. 2005. V. 298. P. 334–364.
Spectral analysis of the anti-reflective transform
A. Arico∗, M. Donatelli† and S. Serra-Capizzano‡
∗Universita degli studi di Cagliari, Italy†‡Universita degli Studi dell’Insubria, Via Ravasi, 2, 21100 Varese, Italy
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∗[email protected], †[email protected],‡[email protected]
Anti-reflective boundary conditions have been studied in connection with fast deblurring algo-rithms, in the case of d-dimensional objects (signals for d = 1, images for d = 2). Here we showhow, under the assumption of strong symmetry of the point spread functions and under mild de-gree conditions, the associated matrices depend on a symbol and define an algebra homomorphism.Furthermore, the eigenvalues can be exhaustively described in terms of samplings of the symboland other related functions, and appropriate O(nd log(n)) arithmetic operations algorithms can bederived for the related computations. These results, in connection with the use of the anti-reflectivetransform, are of interest when employing filtering type procedures for the reconstruction of noisyand blurred objects.
References1. S. Serra-Capizzano, A note on anti-reflective boundary conditions and fast deblurring
models, SIAM J. Sci. Comput., 25 (3) (2003) 1307–1325.
Splitting algorithm for solving variational inequalitieswith inversely strongly monotone operators
I. B. Badriev∗, O. A. ZadvornovKazan State University, Russia
The work was supported by RFBR, grant 06-01-00633∗[email protected]
Let Ω ⊂ Rn, n ≥ 1 be a bounded domain with a Lipschitz continuous boundary Γ. We considerthe following boundary valued problem relatively function u = (u1, u2, . . . , un):
n∑
j=1
(∂
∂xjv(i)j (x) + dij(x)uj(x) ) = fi(x), x ∈ Ω, i = 1, 2, . . . , n, u(x) = 0, x ∈ Γ, (1)
−vj (x) ∈ gj(|∂u(x)/∂xj |)|∂u(x)/∂xj |
∂u(x)
∂xj, x ∈ Ω, j = 1, 2, . . . , n, (2)
where f = (f1, f2, . . . , fn) is given function, D = dij – is a non symmetric matrix such that(Dξ, ξ) ≥ α0 (Dξ, Dξ) for all ξ ∈ Rn, α 0 > 0. We assume that the multi-valued functions gj canbe represented in the form gj(ξ) = g0j(ξ) + ϑj h(ξ − βj), where ϑj, βj are the given non negativeconstants, h is the multi-valued and g0j are the single-valued functions given by the formulas
h(ξ) =
0, ξ < 0,[0, 1], ξ = 0,1, ξ > 0,
g0j (ξ) =
0 , ξ ≤ βj ,g∗j (ξ − βj ), ξ ≥ βj ,
g∗j : [0,+∞)→ [0,+∞) are the continuous functions satisfies the following conditions:
g∗j (0) = 0, g∗j (ξ) > g∗j (ζ) ∀ ξ > ζ ≥ 0, ∃σj > 0 : | g∗j (ξ)− g∗j (ζ) | ≤1
σj| ξ − ζ | ∀ ξ, ζ ≥ 0,
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∃ kj > 0, ξ∗j ≥ 0 : g∗j (ξ∗j ) ≥ kj ξ
∗j , g∗j (ξ)− g∗j (ζ) ≥ kj (ξ − ζ) ∀ ξ ≥ ζ ≥ ξ∗j .
We introduce the next notations: V =[ W
(1)2 (Ω)
]n
, H = [L2(Ω)]n, Bj = ∂/∂xj : V → H,
j = 1, 2, . . . , n. A solution of the problem (1), (2) is defined as the function u ∈ V satisfying for allη ∈ V the variational inequality
(A0u, η − u)V +n∑
j=1
(Aj Bj(u), Bj(η − u))H + F0(η)− F0(u) +n∑
j=1
[Fj(Bjη)− Fj(Bju) ] ≥ 0, (3)
where B∗j : H → V , j = 1, 2, . . . , n, are conjugated to Bj operators, the operators A0 : V → V and
Aj : H → H, j = 1, 2, . . . , n, are generated by the forms
(A0u, η)V =
∫
Ω
(Qu, η)dx, u, η ∈ V, (Aj y, z)H =
∫
Ω
(Gj (y), z)dx, y, z ∈ H,
the operators Gj : Rn → Rn and the functionals F0 : V → R 1, Fj : H → R 1, j = 1, 2, . . . , n, aredefined by the formulas:
Gj (y) = g0j (|y|) |y|−1 y , y 6= 0, Gj (0) = 0, F0(η) = −∫
Ω
(f, η) dx, η ∈ V,
Fj (z) = ϑj
∫
Ω
µ(|z| − βj ) dx, z ∈ H, µ(ζ) =
0, ζ < 0,ζ, ζ ≥ 0.
To find a solution of the variational inequality (3), one can use the following splitting algorithm.
We fix τj > 0, j = 0, 1, 2, . . . , n, and r > 0. Let u(0) ∈ V , y(0)j ∈ H, λ
(0)j ∈ H, j = 0, 1, . . . , n, be
arbitrary elements. For k = 0, 1, 2, . . ., and for known elements y(k)j , λ
(k)j , j = 1, 2, . . . , n, we define
u(k+1) as a solution of the variational inequality
1
τ0
(u(k+1) − u(k), η − u(k+1)
)V
+ F0(η)− F0
(u(k+1)
)+
(A0u
(k), η − u(k+1))
V+
+(
n∑
j=1
B∗j λ
(k)j + r
n∑
j=1
B∗j (Bj u
(k) − y(k)j ) , η − u(k+1) )V ≥ 0 ∀ η ∈ V. (4)
Then we find y(k+1)j , j = 1, 2, . . . , n, from the variational inequalities
1
τj(y
(k+1)j − y(k)
j , z − y(k+1)j )H + Fj(z)− Fj(y
(k+1)j ) + (Aj y
(k)j − λ(k)
j z − y(k+1)j )H+
+r (y(k)j −Bj u
(k+1), z − y(k+1)j )H ≥ 0 ∀ z ∈ H, j = 1, 2, . . . , n. (5)
Finally we set
λ(k+1)j = λ
(k)j + r (Bj u
(k+1) − y(k+1)j ), j = 1, 2, . . . , n. (6)
To analyze the convergence of the method (4)-(6) we formulate it via the transition operatorT : V × H n × H n → V × H n × H n that takes each vector q = ( q0, q1, . . . , q2n ) = (u, Y, Λ ),Y ∈ H n, Λ ∈ H n to the element T q = (T 0 q, T 1 q, . . . , T 2n q ) as follows
T 0 q = Prox τ0 F0( q0 − τ0 [A0 q0 +n∑
j=1
B∗j qn+j + r
n∑
j=1
B∗j (Bj q0 − qj ) ] ) ,
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T jq = Prox τj Fj(qj − τj [Aj qj − qn+j + r (qj −Bj T 0 q ) ] ) , j = 1, 2, . . . , n,
Tn+j q = qn+j + r (Bj T 0 q − T j q ) , j = 1, 2, . . . , n,
where ProxG is the proximal mapping. Then the method (4)-(6) can be represented in the form
q(k+1) = Tq(k), q(k) = (u(k), y(k)1 , y
(k)2 , . . . , y(k)
n , λ(k)1 , λ
(k)2 , . . . , λ(k)
n ), k = 0, 1, 2, . . . , (7)
where q(0) is an arbitrary element.Theorem 1. The point q = (u, y1, y2, . . . , yn, λ1, λ2, . . . , λn) is a fixed point of the T if and only
if yj = Bj u, λj ∈ ∂Fj(yj ) +Aj yj, j = 1, 2, . . . , n; −n∑
j=1
B∗j λ j ∈ ∂F0(u) +A0 u. Moreover, the first
component u of each fixed point of the operator T is a solution of the problem (3).Theorem 2. Let the following conditions τj < (2σj)/(2σj r + 1), j = 0, 1, 2, . . . , n, are hold and
the operator T has at least one fixed point. Then the iterativeq(k)
+∞
k=0constructed by the (7)
converges weakly to q∗ = Tq∗ and ‖ y(k)j −Bj u
(k) ‖H = 0, j = 1, 2, . . . , n as k → +∞.It would be note that for the problem (1), (2) the splitting algorithm (4) – (6) is reduced to
solving of n boundary valued problems with Laplace operators.
Elementary equivalence of unitary linear groups
Balmassov E. S.Moscow State University, Russia
Elementary properties of linear groups were considered in 1961 by A.I.Maltsev. in the paper [1]for the groups GLn(K), SLn(K), PGn(K), PSLn(K), it was proved that they are elementaryequivalent if and only if their dimensions coincide, and the initial fields are elementary equivalent.This theory was continued in 1992, when K. Beidar and A.V. Mikhalev in [2] found the generalapproach to the problems of this type. With the help of logical constructions and model theory theygeneralized Maltsev theorem for the case of the group GL over prime rings with 1/2, over skewfields,and also for some other algebraic systems. Then in 1998 E.I. Bunina in the paper [3] studiedelementary properties of unitary linear groups over fields. There were proved structural theorems onunitary linear groups over fields with involutions with forms of maximal rank, and there were usedonly methods from [1]. Besides, with the help of Beidar and Mikhalev theorems [2] and isomorphismtheorem of I.Z. Golubchik and A.V. Mikhalev [4] in 1998 in the paper [5] E.I. Bunina proved theoremson elementary equivalence of unitary linear groups over associative rings and skewfields with 1/2and 1/3. Now we generalize results of the paper [3] for unitary linear groups over fields, where theform is not necessarily of maximal rank.
Let K be infinite field of characteristics 6= 2 with involution j. Let Q2p+q be the form of rank(2p, q). By U2p+q(K, j,Q) we denote the unitary group of matrices A ∈ GL2p+q(K) such thatAQ2p+qA
∗ = Q2p+q, where A∗ = (Aj)T . We prove the following theorem:Theorem. The groups U2p1+q1(K1, j1, Q2p1+q1) and U2p2+q2(K2, j2, Q2p2+q2), where K1 and K2
are infinite fields of characteristics 6= 2, with involutions j1 and j2, respectively, p1, p2 ≥ 2, q1, q2 ≥ 4,are elementary equivalent if and only if :
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1) p1 = p2, q1 = q2 and the fields K1 and K2 are elementary equivalent as fields with involutions;2) the involution j1 is not identical, 2p1 = q2, 2p2 = q1 and the fields K1 and K2 are elementary
equivalent as the fields with involutions.
References1. Maltsev A.I., On elementary equivalence of linear groups, Problems of mathematics and
mechanics, Novosibirsk, 1961, pp. 110-132 (in Russian).2. Beidar C.I., Michalev A.V., On Malcev’s theorem on elementary equivalence of linear
groups, Contemporary mathematics, 131 (1992), pp. 29–35.3. Bunina E.I. Elementary equivalence of unitary linear groups over fields, Fundamental’naya
i prikladnaya matematika, 4 (1998), pp. 1–14 (in Russian).4. Golubchik I.Z., Mikhalev A.V., Isomorphisms of unitary groups over associative rings,
Zapiski nauchnyh seminarov LOMI AN USSR, 132 (1983), pp. 97–109 (in Russian).5. Bunina E.I., Elementary equivalence of unitary linear groups over rings and skewfields,
Russian Mathematical Surveys, 53(2): 137–138 (1998).
Constructing a Sparse Linear System Hardware Accelerator
J. Vic Batson∗, Oleg V. Diyankov†
∗Sparsix Corporation, 236 West Portal Avenue #221, San Francisco, CA 94127, USA†Sparsix Corporation, Oktyabrsky pr., 19a, 120, Troitsk, Russia
Handling massive amounts of data and complex computations are common aspects of high-performance computing. Sparse linear systems, however, introduce an added level of complexitythat greatly reduces the efficiency of general-purpose computers: non-sequential data access. Mod-ern computer architectures are tuned to access data in sequential blocks. To solve sparse linearsystems, data must often be accessed in a non-sequential manner, leading to massive cache missesand dramatically reducing overall system performance. When analyzing large sparse linear systems,the combination of non-sequential data access and numerous, complex computations can easily causesimulation times to extend into the tens or hundreds of hours, even on powerful general-purpose com-puters.
To address this problem, Sparsix Corporation has developed its Sparse System Accelerator (SSA),a field programmable gate array (FPGA)-based application accelerator designed to significantlyimprove the performance of general-purpose computers when solving large sparse linear systems. TheSSA features two stages of performance improvement: a custom-designed memory subsystem basedon Sparsix’s patent-pending Intelligent Memory Controller (IMC) technology, and a reconfigurablecoprocessor that provides hardware acceleration of certain mathematical functions used in sparselinear systems analysis. This balanced design is unique in that it addresses the issue of non-sequentialdata access as well as the computational challenge.
Sparsix Corporation is an early-stage US-Russia technology firm developing solutions that dra-matically accelerate the speed at which general-purpose computers process sparse linear systems.For more information, email [email protected].
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Averaging operations on matrices
Rajendra BhatiaIndian Statistical InstituteNew Delhi 110016, India
Various notions of means of positive numbers can be extended to positive definite matrices. Ofthese the geometric mean presents interesting difficulties because of noncommutativity of matrixmultiplication and subtleties of matrix order. We will survey different approaches to the problemadvanced by physicists, engineers, and mathematicians.
On ranks of spatial matrices
Maxim BershadskyBen-Gurion University of the Negev, Sapir Academic College
[email protected], [email protected]
The study of spatial matrices dates back to the 19-th century. In particular, the determinants andranks of spatial matrices were defined and the classification of normal forms of 2×2×2 matrices overC and R was obtained. G. Belitskii, V. Sergrichuck and I have recently presented the classificationof m×n×2 - matrices and the list of normal forms for m×2×2 matrices over an arbitrary field K ,based on the Kronecker-Weierstrass theory. In this presentation two- and three-dimensional ranksof spatial matrices are considered and a new invariant with respect to equivalence of spatial matricesover a field K is introduced. The invariant, rA(x) , is an integer-valued function on Kq for a givenm × n × q - spatial matrix A . The function image r(A) :≡ rA(Kq) is particularly connected tothe classical three-dimensional ranks and even provides more information than said ranks. As anapplication of the invariant, the classification of some normal forms of spatial 3 × 3 × 3 matricesare presented.
The Cyclic Reduction Algorithm: From Poisson’s Equation to Stochastic Processesand Beyond
Dario A. BiniDipartimento di Matematica, Universita di Pisa
Cyclic reduction (CR) is a well known algorithm for people working in the area of numericallinear algebra. It was introduced by Gene H. Golub in 1970 for the efficient solution of the Poissonequation in a rectangle and has been adapted and modified to cope with different computationalneeds.
6
In this talk we revisit this celebrated algorithm and describe recent applications in different com-putational frameworks like solving wide classes of queuing models, more general stochastic processes,and matrix equations of different kind like unilateral polynomial equations and the Riccati equation.
We discuss the functional interpretation of CR which allows one to relate this algorithm withthe Dandelin-Graeffe-Lobachevsky iteration, to analyze its convergence properties and to extend itto solving (generalized) block Hessenberg block Toeplitz systems.
Finally, we report the results of a recent research which aims to relate CR with the StructuredDoubling Algorithm (SDA) introduced to solve algebraic Riccati equations.
Multi-Level Partitioning Algorithms
N. Bochkarev∗, O. Diyankov∗, V. Pravilnikov∗,B.Beckner†, A. Usadi†, I. Mishev†, S. Maliassov†
∗Neurok TechSoft LLC, Troitsk, Moscow reg., Russia†ExxonMobil Upstream Research Company, Houston, TX, USA
The presentation describes a multi-level approach on the construction of effective partitioning ofunstructured graphs used in parallel computations. The quality of partitioning is estimated by itsusing in parallel iterative solution of large sparse linear systems arising in discretization of PDE onunstructured grids. Various algorithms of coarsening, and refinement, and balancing under certainconstrains are considered and numerically compared on a set of test matrices.
Analytic method for computation of magnetic field in complex domain
A.BogatyrevInstitute of Numerical Mathematics of RAS
The minimization of hard magnetic discs requires the highest density of recorded data. This inturn brings about the necessity of high accuracy (7-8 true digits) computation of magnetic field. Theproblem in 2D domain bounded by the soft underlayer, two shields and the reading magnetoresistivehead is solved analytically in terms of elliptic theta functions. The proposed method allows to solveboundary value problems for harmonic functions with extreme precision in other domains as well.
An approach to solution of nonlinear Helmholtz eigenvalue problems stemming fromintegrated optics with Jacobi-Davidson solvers
7
Mikhail BotchevUniversity of Twente, the Netherlands
We consider nonlinear, nonpolynomial eigenvalue problems stemming from finite element dis-cretization of the Helmholtz equation and arising in simulation of integrated optical devices. TheHelmholtz equation is originally posed in an infinite domain but solved numerically in a boundeddomain with artificial boundary conditions. These so-called nonreflecting or transparent bound-ary conditions should guarantee transparency of the domain boundary for outgoing waves. Thenonreflecting boundary conditions used in this work are the recently developed transparent-influxboundary conditions [Nicolau & Van Groesen, 2005]. These conditions are obtained by solving theproblem in the exterior of the computational domain analytically and have a number of advantagesas compared to other known formulations of transparent boundary conditions.
A finite element discretization of the Helmholtz equation leads to an eigenvalue problem wherethe transparent-influx boundary conditions cause a nonlinear, nonpolynomial dependence of thematrix on the eigenvalue. Moreover, it is not trivial to express this nonlinearity in an explicit way.However, since the nonlinearity results from the boundary conditions, the nonlinear contributionsto the matrix of the eigenvalue problem can be seen a smaller dimensional discrete operator. Thisallows for a relatively cheap low-rank parametrization of the nonlinear dependence so that it canbe approximated by a low-degree matrix polynomial. Thus, we reduce the nonlinear nonpolynomialeigenvalue problem to a nonlinear polynomial one. Once this reduction is done, the Jacobi-Davidsonmethod can readily applied. Depending on the accuracy requirements of the eigenvalue problem,the polynomial approximation can be refined during the Jacobi-Davidson iterations.
This research is based on a joint work in progress with Ardhasena Sopaheluwakan, Gerard Slei-jpen, Brenny van Groesen, Jaap van der Vegt and Manfred Hammer.
Automorphisms of adjoint Chevalley groups over local rings
E. I. BuninaDepartment of Mechanics and Mathematics, Moscow State University, 119992, Moscow, Russia
Suppose that Gad is a Chevalley-Demazure groups scheme associated with irreducible root sys-tem Φ; Gad(R,Φ) is the group of points Gad with values in R; Ead(R,Φ) is an elementary subgroupin Gad(R,Φ), where R is a local commutative ring with 1. We describe automorphisms of the groupsGad(R,Φ) and Ead(R,Φ) in the case when R is a local ring with 1/2, with such an invertible el-ement α, that α2 − 1 is also invertible, and if the root system has the type G2, then 1/3 ∈ R,and we suppose that the root system has the rank > 1. Then every automorphism of Gad(R,Φ)(Ead(R,Φ)) is standard (see below). These results for Chevalley groups over fields were obtained byR. Steinberg [1] for the finite case and by J. Humphreys [2] for the infinite one. E. Abe [3] provedthis result for Noetherian rings, but the class of local rings is not contained completely in the classof Noetherian rings, and proofs of the paper [3] can not be extended for arbitrary local rings. Weuse some methods from [4].
Let us define 4 types of automorphisms of a Chevalley group Gπ(R,Φ), that are called standard.Suppose that CG(R) is the center of Gπ(R,Φ) and τ : Gπ(R,Φ)→ CG(R) is a group homomorphism.
8
Then the mapping x 7→ τ(x)x from Gπ(R,Φ) onto itself is an automorphism of Gπ(R,Φ), that iscalled a central automorphism of Gπ(R,Φ). Let ρ : R→ R be an automorphism of the ring R. Themapping x 7→ ρ x from Gπ(R,Φ) onto itself is an automorphism of Gπ(R,Φ), that is called a ringautomorphism of Gπ(R,Φ). Let V be the representation space of the group Gπ(R,Φ), and supposethat g ∈ GL(V ) is a matrix such that gGπ(R,Φ)g−1 = Gπ(R,Φ). Then the mapping x 7→ gxg−1 fromGπ(R,Φ) onto itself is an automorphism of G(R), that is called the inner automorphism of G(R),induced by the element g from GL(V ). Suppose that ∆ is a basis of Φ (that is fixed), and δ is anautomorphism of Φ such that δ∆ = ∆. Then there exists a unique automorphism Gπ(R,Φ) suchthat for every α ∈ Φ and t ∈ R the element xα(t) is mapped to xδ(α)(ε(α)t), where ε(α) = ±1 for allα ∈ Φ and ε(α) = 1 for all α ∈ ∆. It is called a diagram automorphism of Gπ(R,Φ). Similarly wecan define 4 types of automorphisms of the elementary subgroup E(R). An automorphism σ of thegroup Gπ(R,Φ) (or Eπ(R,Φ)) is called standard, if it is a composition of automorphisms of these 4types.
Theorem. Suppose that Ead(R,Φ) is an elementary Chevalley group with irreducible root systemof rank > 1, R is a commutative local ring with 1/2. Suppose that R contains an invertible element αsuch that α2 − 1 is invertible in R, and if the root system Φ has the type G2, then 1/3 ∈ R. Thenevery automorphism of Ead(R,Φ) is standard.
There is the similar theorem for Chevalley groups Gad(R,Φ).
References1. Steinberg R., Automorphisms of finite linear groups, Canad. J. Math., 121, 1960, 606–615.2. Humphreys J. F., On the automorphisms of infinite Chevalley groups, Canad. J. Math., 21,
1969, 908-911.3. Abe E. Automorphisms of Chevalley groups over commutative rings, Algebra and Analysis,
5(2), 1993, 74–90.4. Petechuk V.M. Automorphisms of groups SLn, GLn over some local rings, Mathematical
Notes, 28(2), 1980, 187–206 (in Russian).
Mirsky-based tools for the eigenvalue distribution in a non-Hermitian setting
Stefano Serra-CapizzanoUniversita degli Studi dell’Insubria, Via Ravasi, 2, 21100 Varese, Italy
The work of the author is supported in part by MIUR grant no. [email protected]
Under mild trace norm assumptions on the perturbing sequence, we extend a recent Mirski-basedperturbation result concerning the eigenvalues of a generic (non Hermitian) complex perturbation of abounded Hermitian sequence of matrices. Some examples of application are considered, ranging fromthe algebra generated by Toeplitz sequences to the approximation of PDEs with various boundaryconditions. A final discussion on open questions and further lines of research ends the note.
References1. R.Bhatia, Matrix Analysis, Springer Verlag, New York, 1997.2. R.Bhatia, Pinching, trimming, truncating, and averaging of matrices, Amer. Math. Monthly,
107 (2000), 602–608.
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3. D.Fasino and S. Serra-Capizzano, From Toeplitz matrix sequences to zero distribution oforthogonal polynomials, Contemp. Math. 323 (2003), 329–340.
4. L.Golinskii and S. Serra-Capizzano, The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences, J. Approx. Theory, 144-1 (2007), pp. 84–102.
5. S.Holmgren, S. Serra-Capizzano, and P. Sundqvist, Can one hear the composition ofa drum?, Mediterr. J. Math., 3-2 (2006), pp. 227–249.
6. A.B. J.Kuijlaars and S. Serra-Capizzano, Asymptotic zero distribution of orthogonalpolynomials with discontinuously varying recurrence coefficients, J. Approx. Theory 113 (2001),142–155.
7. S. Serra-Capizzano, Generalized Locally Toeplitz sequences: spectral analysis and applica-tions to discretized Partial Differential Equations, Linear Algebra Appl. 366-1 (2003), 371–402.
8. S. Serra-Capizzano, The GLT class as a Generalized Fourier Analysis and applications,Linear Algebra Appl., 419-1 (2006), pp. 180–233.
9. S. Serra-Capizzano, D. Bertaccini and G.H.Golub, How to deduce a proper eigenvaluecluster from a proper singular value cluster in the non normal case, SIAM J. Matrix Anal. Appl.,27-1 (2005), pp. 82–86.
10. S. Serra-Capizzano and P. Sundqvist, Stability of the notion of approximating class ofsequences and applications, J. Comput. Appl. Math., in press.
11. P.Tilli, A note on the spectral distribution of Toeplitz matrices, Linear Multilin. Algebra,45 (1998), 147–159.
12. P.Tilli, Some results on complex Toeplitz eigenvalues, J. Math. Anal. Appl. 239-2 (1999),390–401.
13. E.Tyrtyshnikov and N. Zamarashkin, Spectra of multilevel Toeplitz matrices: advancedtheory via simple matrix relationships, Linear Algebra Appl., 270 (1998), pp. 15–27.
p-Sparse BEM for weakly singular integral equationon random boundary
Alexey Chernov∗ and Christoph Schwab†
∗†Seminar for Applied Mathematics, ETH Zurich, 8092 Zurich, Switzerland∗[email protected], †[email protected]
We consider the weakly singular boundary integral equation Vu(ω) = g on a randomly perturbedsmooth closed surface Γ(ω) (cf. [2] for stochastic g). The aim is the computation of the momentsMku := E[⊗k
i=1u], k ≥ 1, if the corresponding moments of the perturbation are known. The problemon the stochastic surface is reduced to a problem on the nominal deterministic surface Γ with therandom perturbation parameter δβ(ω). Note, that u(ω) depends nonlinearly on δβ(ω).
Resulting formulation for the kth moment is posed in the tensor product Sobolev spaces andinvolve the k-fold tensor product operators V(k) = ⊗k
i=1V. The complexity of the standard fulltensor product Galerkin BEM is O(Nk), where N is the number of degrees of freedom needed todiscretize the nominal surface Γ. Based on [3], we develop the p-sparse grid Galerkin BEM to reducethe problem complexity to O(N(logN)k−1).
References
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1. H. Harbrecht, R. Schneider, Ch. Schwab, Sparse second moment analysis for ellipticproblems in stochastic domains, in press.
2. T. von Petersdorff, Ch. Schwab, Sparse finite element methods for operator equationswith stochastic data, Appl. Math., 51 (2006), no. 2, 145–180.
3. V. N. Temlyakov, Approximation of periodic functions, Nova Science Publ., New York,1994.
Continuation method to solve symmetric indefinite Toeplitz systems
Andrey A. Chesnokov∗†, Marc Van Barel†∗Dept. Comp. Math. and Cyb., Moscow State University;
†Dept. Comp. Sci., Katholieke Universiteit Leuvenandrey.chesnokov, [email protected]
The method being presented starts with the identity matrix and gradually transforms it to theinverse of an indefinite Toeplitz matrix. If the Toeplitz matrix is ill-conditioned, the method performsseveral Newton iterations at the very end. Overall memory requirement is of the order O(n). Thecomputational complexity per iteration step is of the order O(n log n).
The validity of the approach is illustrated by numerical experiments.
On a general approach to deriving the known classesof normal Hankel matrices
Chugunov V. N.∗, Ikramov Kh.D.†∗Institute of Numerical Mathematics of RAS
†Moscow State University, Mechanics and Mathematics Dept.∗[email protected], †[email protected]
The problem of characterizing normal Hankel matrices is still far from being completely solved.The available partial results are mainly the descriptions of specific subsets of this matrix class. Thesesubsets were found by considerations that are specific for each subset. In this talk, we present ageneral approach that allows us to obtain all the known subsets as special cases of a unified scheme.Normal Hankel matrices outside of these subsets correspond to the main (and the most difficult) caseof our scheme. This case, which at the moment does not yield to a complete analysis, gives originto a new and very interesting matrix class that extends (and contains) the class of ϕ-circulants.
Decompositions of a Higher-Order Tensor in Block Terms
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Lieven De LathauwerCentre National de la Recherche Scientifique Lab. ETIS (ENSEA, UCP, CNRS UMR 8051) 6,
avenue du Ponceau, BP 44, F 95014, Cergy-Pontoise Cedex, [email protected]
In this talk we introduce a new class of tensor decompositions. Intuitively, we decompose a giventensor “block” into “blocks of smaller size”, where the size is characterized by a set of mode-n ranks.We study different types of such decompositions. For each type we present conditions under whichessential uniqueness is guaranteed. The Parallel Factor decomposition and Tucker’s decompositioncan be considered as special cases in the new framework. The results shed new light on fundamentalaspects of tensor algebra.
Development of parallel iterative algorithms for the solution of 2D elasto-plasticproblems
Elena N. Akimova∗, Irina P. Demeshko†, Anatolii V. Konovalov†
∗Institute of Mathematics and Mechanics of UrB RAS, S. Kovalevskaya str.,16, Ekaterinburg,†Institute of Engineering Science of UrB RAS, Komsomolskaya str.,34, Ekaterinburg
∗[email protected], †[email protected]
In this work we investigate the applicability of parallel computations for the solution of elasto-plastic problems with small elastic and large plastic strains using finite element method.
As an example, we consider the problem of cylinder compression from elasto-plastic isotropicand isotropic-strengthened material by flat plates. The solution is based on a principle of virtualcapacity of the speed form
∫
V
(σ + ∆tσ) · ·∇hdV +
∫
Σ
(P + ∆tP ) · hdΣ = 0. (1)
Here σ — is Cauchy’s strain tensor; P — surface force density; ∆t — period for an incrementinterval of load; h and ∇h — kinematically admissible velocity fields variation and its inverteddelta; V, Σ — solid volume and surface; dV, dΣ — volume and cylinder surface area elements.These defining equations for the strain tensor velocity are taken from the work [1].
On contact with plates the Coulomb’s law was applied. A lateral surface of the cylinder is freefrom loadings. By means of finite-element approximation the equation (1) is reduced to the linearalgebraic equations system (LAES)
Az = b, (2)
A, b, z — initial matrix, right part vector and decision vector of the system, respectively. Here A isthe asymmetrical banded matrix.
Solving the problem (1) by means of Finite Element Analysis consist of 3 basic stages: making ofthe matrix A, solving of the LAES (2), calculating of deflected mode at the end of loading iteration.
Parallel algorithms for solving the first and third stages were developed. This stages are easy forparallelization, but solving time of its are about half past of the problem time. The rest of time isspent for solving of the linear algebraic equations system. Time results of that algorithms showed that
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with increasing number of processor a paralleling effectiveness is increase. However, with increasingmatrix dimension a paralleling effectiveness reduces for every examined iterative method because ofa send data time is increase. The comparison of computational efficiency of parallel algorithms at thecommon memory multiprocessor computer with the time at the distributed memory multiprocessorcomputer are performed. The effectiveness of the solving at the common memory multiprocessorcomputer higher that at the distributed memory multiprocessor computer because of send data timehigher at memory multiprocessor computer
For the solving of the linear algebraic equations system a direct method and a iteration methodsare reviewed. As a direct method the Gauss method was applied, but its parallelization [2] didn’tshow the reduction of the calculation time due to the matrix being banded. The parallel algorithmswere developed for the following iterative methods [3]: the Prime Iteration method (MPI), theMinimal residual method (MMR), the Steepest descent method (MSD) and the Method of conjugategradients (MCG).
Parallelization algorithms is based on transformation of a banded matrix into a vertical strip andbanding it on m blocks, and the A — parent matrix, right part vector and decision vector of thesystem, accordingly vector of solving and the right part vector of the linear algebraic into m partsso, that n = m ·L, where n — dimension of the equations system, m — number of processors. Eachprocessor calculates a part of the solving vector on each iteration.
Numerical realization and parallelization of the iterative methods for solving the linear algebraicequations has been implemented on the Massively Parallel Computing System MVS-1000/32 with32 processors with help of the MPI C++ program library.
Solving results of the problem (2) by the parallel iterative algorithms for the banded matrix thewidth of which is 171 and the number of variables is 3362 are shown in the picture. It is evidentthat the Prime Iteration method, the Minimal residual method and the Steepest descent methodshow the same results. The method of conjugate gradients requires to do more iteration than theother examined methods and needs more time for solving the linear algebraic equations.
Fig 1. The LAES executing times by the parallel iterative algorithms
With increasing matrix dimension a paralleling effectiveness rises for every examined iterativemethod. This happens because the increase of matrix dimensions leads to the increase of thelaboriousness of each matrix parts, located on each processor. At the same time, data message levelincreases slightly. As a result, computation time: time of transmission increases rises, when matrixdimension increase. So, with the increasing matrix dimension we can obtain larger computationalspeedup at the larger processor number, if we use a parallel iterative algorithms. Also, comparison
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of computational speedup and efficiency of parallel algorithms and comparison the computing timewith the computing time of the Gauss method were performed.
The obtained results show, that the computing time of linear algebraic equations by the Gaussmethod on one processor is less than the computing time of this system by the iterative methods onone processor for moderate matrix. But, in comparison with the Gauss method, the iterative methodshave higher efficiency of parallelization. The use of parallel algorithms reduces the computing time,that allows to recommend them for the solving of greater elasto-plastic problems.
References1. Konovalov A.V. Constitutive equations for elasto-plastic medium with large plastic defor-
mations, Proceedings of the RAS. Mechanics of rigid body. 1997. V. 5. P. 139–149.2. Korneev V.D. Parallel programming by MPI, Novosibirsk, The Russian Academy of Sci-
ences. 2000. 213 p.3. Fadeev V.K., Fadeeva V.N. Calculating methods of the linear algebra, Moscov: State
Publishing House of the physicotechnical literature. 1963. 734 p.
Simple wavelets for irregular grid
Yu.K.Dem’yanovichSaint-Petersburg State University, Russia
Research is supported in part by RFFI grants 04-01-00269 and [email protected]
Excelent results of wavelet theory and their applications are obtained with powerful Fourier’sapparatus (for example, see [1], [2]). The proposed work deals with an algebraic approach whichleads to simple spline-wavelet decompositons.
Let Xr, r = 0, 1, 2, . . ., be a system of irregular grids in interval (α, β) ⊂ R1,
Xr : . . . < x(r)−1 < x
(r)0 < x
(r)1 < . . . ; lim
j→−∞x
(r)j = α, lim
j→+∞x
(r)j = β,
with embedding property X0 ⊂ X1 ⊂ X2 ⊂ . . .. We discuss a vector-functionϕ(t)
def= (ϕ0(t), ϕ1(t), . . . , ϕm(t))T , t ∈ (α, β), with measurable linear independent
components ϕk(t), k = 0, 1, . . . ,m.A system of embedded (general speaking, non-polynomial) spline spaces S(Ar,ϕ) ⊂ S(Ar+1,ϕ),
r = 0, 1, 2, . . . , is constructed, and decomposition of the system into direct sum of wavelet spacesS(Ar+1,ϕ) = S(Ar,ϕ)
.+W r is realized. Sequential usage of the decomposition leads to represantation
of space Sdef=
⋃+∞r=0 S(Ar ,ϕ) with direct sum,
S = S(A0,ϕ).+W
0 .+W
1 .+W
2 .+ . . . . (1)
As a result, simple formulars of decomposition and reconstruction are obtained. The space S(A,ϕ)
is defined with vector-function ϕ and with complete chain Adef= mkk∈Z of points mk
def= (xk, pk) of
direct product Mdef= (α, β) × P m of the interval (α, β) and m-dimensional projective space P m. An
involution operator (here it is signed by symbol ∗), which gives local orthogonal chain, is introduced
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in the set of complete chains; then embedding of mentioned spaces can be written in the formS(A,ϕ) ⊂ S(B∗,ϕ), where B is a complete extension of the chain A∗ (notice that set of completeextensions has the power of the continuum). The wavelet decomposition (1) is derived with caliberrelations (generalizing the multi-scaled equation) and with a functional system which is biorthogonalto the system of basic functions in the space S(Ar,ϕ) for each r = 0, 1, 2, . . .. Basic wavelets of theproposed decompositions possess compact supports (of length of m+1 grid intervals), and they havesimple analytical expression. Under different assumptions as to vector-function ϕ(t) the necessaryand sufficient conditions of belonging of the wavelets to spaces W s
q and Cs are formulated. Theorder of approximation with discussed wavelet spaces are asymptotically optimal with respect toN -width of standard compact sets. If the grid X and the vector-function ϕ ∈ Cm−1(α, β) are fixedthen the space S(A,ϕ) which belongs to Cm−1(α, β) is unique among discussed spline spaces; thisspace is called the space of Bϕ-splines of order m (see [3]). Some examples are discussed.
References1. Charles K. Chui, An Introduction to Wavelets, Academic Press, 1992.2. S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999.3. Yu. K. Dem’yanovich, Smoothness of spline spaces and wavelet decompositions, Dokl.
Ross. Akad. Nauk 401 (2005), no. 4, pp. 439–442.
Solving Systems with Hierarchical Structure
Patrick DewildeTU Delft, the [email protected]
The holy grail of solving large systems of equations is to do it (in a stable way) with a complexityequal to the number of independent parameters determining the system (the algebraic degree offreedom). Although in most cases it appears to be next to impossible to achieve this feat, we dodispose of techniques that are capable of handling some systems with structure efficiently, suchas some types of sparse systems, semi-separable systems, Toeplitz or Hankel systems and evencombinations of those. A new challenge is to handle systems with hierarchical structure. We shallconcentrate on systems consisting of a tree of subsystems which again may have structure at a lowerlevel of the hierarchy. We shall connect with some classical results on the Toeplitz block Toeplitz case,consider the so called HSS (Hierarchically Semi Separable) systems pioneered by Chandrasekharanand Gu, look at the invariant multi-resolution case of Alpay and Volok, and derive a methodologyand conditions under which solutions can generally be constructed.
Fuzzy Grid Method for Numerical Solution of PDE
Oleg V. DiyankovNeurok Techsoft LLC, office 120, 19a Oktyabrskiy ave., Troitsk, Moscow region, Russia, 142190
15
The Fuzzy Grid Method (FGM) for PDE solution is presented. The method is a natural gener-alization of meshless methods, based on Free-points method and SPH methods ideas. The methodis illustrated with numerical samples of 2D Gas Dynamics equations and Poisson equation solution.
There are different approaches to construction of numerical methods for PDE solution. The mostpopular are: Finite Difference (FD), Finite Element (FE), Particle in Cell (PIC), Free Lagrange(FL). All of these methods have their own advantages and drawbacks. The paper is devoted to theintroduction of new approach, developing the two mentioned above meshless approaches (PIC andFL), which inherits the advantages of meshless approaches allowing modeling of large deformationsand overcomes drawbacks, first of all linked with nonmonotonicity of the received numerical solutions.
The main idea of the method is based on the approximation of the surface and volume integralsof the unknown functions in a particle as sums of these functions values in the points (particles),which are in the nearest vicinity to the given one.
∫V
fdV |i ≈∑
j∈Ωi
Ai,j · fj
∮∂V
fdS|i ≈∑
j∈Ωi
Bi,j · fi,j(1)
Here Ωi is a set of numbers of neighbor particles for the particle number i, fi,j — approximationof function f at the “fuzzy face” between i-th and j-th particles.
The coefficients in (1) should satisfy some relations. So, for each particle we have somethinglike a cloud of particles around it, which is used for the approximation construction. The name“particle” is used as a close to reality one, but really we have a volume, which doesn’t have strictbounds, but about which we know values, which are necessary for modeling of a specific appliedproblem. That is why, this method was called “Fuzzy Grid”.
Monotone transformations with respect to♯6 and
cn
6-partial orders
Mikhail A. Efimov[ poster presentation ]
Lomonosov Moscow State [email protected]
Let Mn(F) be an algebra of n× n-matrices with entries from an arbitrary field F. A generalizedinverse matrix A− for a fixed matrix A ∈ Mn(F) is defined to be any solution of the matrixequation AA−A = A. A generalized inverse matrix A−
r for the matrix A, which in addition satisfiesthe condition A−
r AA−r = A−
r , is called a reflexive generalized inverse. A group generalized inversematrix A♯ (or simply group inverse ) for the matrix A is defined to be a reflexive generalized inversematrix which commutes with the matrix A. A matrix A is said to be of index 1 if rkA2 = rkA.
In this talk we deal with partial orders which can be defined in terms of group inverse. Theirstudying is important today in connection with different aspects of general algebra and mathematicalstatistics.
Definition 1. [4] Let A ∈Mn(F) be a matrix of index 1 and B ∈Mn(F) be an arbitrary matrix.
We say that A♯6 B iff AA♯ = BA♯ = A♯B.
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We note, that♯6 is rather poor, namely, all matrices of index bigger than one are maximal.
R. Hartwig and S.-K. Mitra in [3] introduced the order relation on the whole matrix algebra Mn(F)
which extends♯6-order. This relation is called
cn6-order. To define this order we need to introduce the
following decomposition for an arbitrary matrix A ∈Mn(F). Let F be an arbitrary field. The core-nilpotent decomposition of a square matrix A ∈Mn(F ) is the following decomposition: A = CA+NA,where NA is nilpotent matrix and CA is a matrix of index 1, moreover CANA = NACA = 0. Thisdecomposition is unique and exists for any square matrix A, see [1, Chapter 4.8]. The other notion,we need, is the following well-known notion of minus-order : A6B iff rk (B −A) = rkB − rkA.
Definition 2. [3] Let F be an arbitrary field. For a pair of matrices A,B ∈Mn(F) we have:
Acn6 B, iff
CA
♯6 CB
NA 6 NB
We investigate transformations on matrix algebra, which preserve♯6 and
cn6-partial orders. In [2]
linear bijective transformations preserving each of these orders are characterized. Here we removethe assumptions of bijectivity and show that linearity can be substituted by additivity. Namely, weprove the following:
Theorem 1. Let F be a field with |F | > 3, n be a positive integer, n > 2. Let, moreover, additive
mapping T : Mn(F) → Mn(F) is monotone with respect to♯6 or
cn
6-partial orders. Then thereexist α ∈ F, P ∈ GLn(F) and ϕ ∈ AutF, such that T has the form T (X) = αP−1ϕ(X)P for allX ∈Mn(F) or T (X) = αP−1ϕ(X)tP for all X ∈Mn(F), where Xt denotes transposed matrix.
I would like to thank my scientific advisor Dr. Alexander E. Guterman for posing this problemand fruitful discussions.
References1. Ben-Israel A., Greville T., Generalized Inverses: Theory and Applications, New York,
John Wiley and Sons, 1974.2. Bogdanov I. I., Guterman A. E. Monotone matrix transformations defined by the group
inverse and simultaneous diagonalizability, Matematicheskii Sbornik, 198(1), 2007, 3–20.3. Hartwig R. E., Mitra S. K. Partial orders based on outer inverses, Linear Algebra Appl.,
176 (1982), 3–20.4. Mitra S. K. On group inverses and the sharp order, Linear Algebra Appl. 92 (1987), 17–37.
Efficient generators of quasiseparable matrices
Yuli Eidelman∗ and Israel GohbergTel Aviv University, Israel∗[email protected]
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We study a class of block structured matrices R = RijNi,j=1 whose entries are specified asfollows:
Rij =
piai−1 · · · aj+1qj, 1 ≤ j < i ≤ N,di, 1 ≤ i = j ≤ N,gibi+1 · · · bj−1hj , 1 ≤ i < j ≤ N.
Here pi, qj, ak are matrices of sizes mi × r′i−1, r′j ×mj , r
′k × r′k−1 respectively; these elements are
said to be lower generators of the matrix R with orders r′k. The elements gi, hj , bk are matrices ofsizes mi × r′′i , r′′j−1 ×mj, r
′′k−1 × r′′k respectively; these elements are said to be upper generators of
the matrix R with orders r′′k . The diagonal entries dk are matrices of sizes mk ×mk.Let m = max1≤k≤N mk be the maximal size of blocks of R. We assume that m is a small number.
Let r be the maximal order of generators of the matrix R:
r = max max1≤k≤N−1
r′k, max1≤k≤N−1
r′′k.
In terms of lower and upper generators and diagonal entries of a quasiseparable matrix one canobtain O(r2N) algorithms to compute the product y = Rx and O(r3N) algorithms to solve thesystem of linear algebraic equations Rx = y. In this talk we discuss a special type of generatorsfor which the multiplication of a quasiseparable matrix by a vector and solution of a quasiseparablesystem may be done in O(rN) and O(r2N) operations respectively. We show that for any matrixsuch efficient generators exist. Various algorithms to compute these generators are presented.
A use of matrix methods for solution and stability analysis in convective problems
Michael ErmakovInstitute for Problems in Mechanics RAS, Moscow, Russia
An experience of a use of matrix methods for a solution of axisymmetric convective problemson the basis of Navier-Stokes equations for a viscous heat-conductive non-compressible fluid is de-scribed. The governing equations consist of continuity, momentum and energy conservation equa-tions. Approximation of governing equations in natural variables (velocity, pressure) is effected byfinite volume method on staggered mesh. The boundary conditions and relations for “fictious” pointsare included in the joint system. The system of non-linear equations is solved by Newton-Raphsonmethod. The solution of sparse linear system for Jacobi matrix is effected by stabilized bi-conjugategradient (BiCGStab) method with ILU-preconditionning [1]. An advantage of the generalized resid-ual method (GMRes) against BiCGStab is not recovered in our problems. The dimension of solvedproblems are up to 200000 and the calculation time is about 10 minutes. The effective solution ofhigher dimension problem is restricted by huge sizes of prenditionners.
For a linear stability analysis a representation of perturbations at the same discrete mesh asa basic solution is used, in an azimuthal direction the solution is presented as a superposition ofinteger-valued normal modes. The solution of generalized eigen-value problem is effected by inverseiteration method. For solution of a linear system the version of BiCGStab and ILU factorizationfor sparse complex matrices. The time of obtaining of a pair eigen vector — eigen value for 200000dimension vector is about 20 miutes.
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The advantages of matrix methods are in a relative simplicity of a numerical code developmentand a high numerical stability of created algorithms. A use of linear stability analysis allows toaccelerate essentially, in compare to direct numerical modeling , a search of critical conditions. Theproposed procedure of a matrix solution of Navier-Stokes equations and a linear stability analysisare carefully tested [2-4]. The applications of this research are a simplified models of crystal growth:half-zone model and Czochraski model [2-4].
An efficiency of existing ILU-preconditionners, a development of optimal preconditionners and aperspectives of 3-D fluid modeling are also discussed in the paper.
References1. Y. Saad, Iterative methods for sparse linear systems, 2nd edition, SIAM, 2003.2. M. Ermakov, Linear Stability Analysis of Axisymmetric Convective Flows, AIAA 2006-3793.3. M.K. Ermakov, M.S. Ermakova, Linear-stability analysis of thermocapillary convection
in liquid bridges with highly deformed free surface, J. Crystal Growth 266 (2004) 160–166.4. V. Shevtsova, Thermal convection in liquid bridge with curved free surfaces: benchmark on
numerical solutions, J. Crystal Growth 280 (2005) 632–651.5. W. Shi, M.K. Ermakov, N. Imaishi, Effect of pool rotation on thermocapillary convection
in shallow annular pool of silicone oil, J. Crystal Growth 294 (2006) 474–485.
Superoptimal approximation for ill-conditioned linear systemswith unbounded symbols
C. Estatico∗ and S. Serra Capizzano†
∗Dipartimento di Matematica e Informatica, Via Ospedale 72, 09124 Cagliari (ITALY),†Dipartimento di Fisica e Matematica, Via Valleggio 11, 22100 Como (ITALY)
∗[email protected], †[email protected], †[email protected]
In this talk, we discuss some properties of the superoptimal Frobenius approximation for thesolution of ill-conditioned linear systems with Toeplitz matrices generated by unbounded symbols.In particular, we use the superoptimal approximation as preconditioner for the CG method whena Fisher-Hartwig singularity is present in the symbol, with special regard to systems coming fromtimes series and financial applications.
The idea behind the superoptimal approximation is to approximate the system matrix by consid-ering an optimization procedure concerning a kind of relative error in the matrix sense, instead of theabsolute error considered in the classical optimal preconditioning. On these grounds, we comparethe very different behavior between (i) the relative approximation of the generating function and(ii) the relative approximation of the generated Toeplitz matrix.
A theoretical discussion concerning classical circulant preconditioners and a numerical compar-ison with the Strang and with the optimal approximations are also presented particularly withreference to the presence of noise.
References1. F. Di Benedetto, C. Estatico, S. Serra-Capizzano, Superoptimal preconditioned con-
jugate gradient iteration for image deblurring, SIAM J. Sci. Comput., 26-3 (2005), pp. 1012–1035.
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2. F. Di Benedetto, S. Serra-Capizzano, A note on the superoptimal matrix algebra oper-ators, Linear Multilin. Algebra, 50 (2002), pp. 343–372.
3. C. Hurvich, Y. Lu, On the complexity of the Preconditioned Conjugate Gradient algorithmfor solving Toeplitz systems with a Fisher-Hartwig singularity, SIAM J. Matrix Anal. Appl., 27(2005), pp. 638–653.
4. E. Tyrtyshnikov, Optimal and superoptimal circulant preconditioners, SIAM J. MatrixAnal. Appl., 13 (1992), pp. 459–473.
Tensor product approximation in quantum chemistry
Heinz-Jurgen FladInstitut fur Informatik, Christian-Albrechts-Universitat zu Kiel
Tensor products provide a versatile tool to approximate various quantities encountered in elec-tronic structure calculations. The talk provides an overview on some recent results concerning thebest N -term approximation of these quantities in anisotropic wavelet tensor product bases. Fur-thermore, so-called best separable approximations have been studied which do not rely on a fixedmultivariate basis anymore. In combination with stable quadrature schemes for the Coulomb in-teraction, these tensor products enable an efficient evaluation of two-electron integrals. It will beshown that substantial further compressions with respect to the tensor product rank can be achievedcompared to traditional Gaussian-type basis sets.
Non-linear commutativity preserving maps on symmetric matrices
Ajda FosnerUniversity of Maribor, Faculty of Natural Sciences and Mathematics, Koroska cesta 160,
2000 Maribor, [email protected]
Let Sn(C) be the set of all symmetric matrices in Mn(C), n ≥ 3. We will study bijective mapson the set Sn(C) that preserve commutativity in both directions. We will show that if φ : Sn(C)→Sn(C), n ≥ 3, is a continuous bijective map preserving commutativity in both directions, then thereexist an orthogonal matrix Q ∈ Mn(C) and for every A ∈ Sn(C) a complex polynomial pA suchthat either φ(A) = QpA(A)Qt, A ∈ Sn(C), or φ(A) = QpA(A)Qt, A ∈ Sn(C). We will also studycommutativity preserving maps on Sn(C) without the continuity assumption.
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On superderivations and local superderivations
Ajda Fosner and Maja FosnerUniversity of Maribor, Faculty of Natural Sciences and Mathematics, Koroska cesta 160,
2000 Maribor, [email protected]
We describe superderivations in certain superalgebras by their actions on elements satisfyingsome special conditions. One of the main results is applied to local superderivations on some certainsuperalgebras.
Computing Zeros of Polynomials
Walter GanderETH Zurich
Jim Wilkinson discovered that the computation of zeros of polynomials is ill conditioned whenthe polynomial is given by its coefficients. We review and demonstrate this fact. For many problemswe need to compute zeros of polynomials, but we show that we do not necessarily need the coefficientsof the polynomials to solve the problem. We develop algorithms that avoid the coefficients. Theyturn out to be stable, however, the drawback is often heavily increased computational effort. Onthe other hand modern processors are mostly idle and wait for crunching numbers so it may pay toaccept more computations in order to increase stability. We show and demonstrate examples for thequadratic eigenvalue problem and for orthogonal polynomials.
Linear projectors on tropical spaces
G. Cohen∗, S. Gaubert†§, J.-P. Quadrat‡§∗ENPC 6-8, avenue Blaise Pascal Cite Descartes, Champs-sur-Marne 77455 Marne-La-Vallee
Cedex 2, France§INRIA-Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France
∗[email protected], †[email protected], ‡[email protected]
The max-plus semiring consists of the set R∪−∞ equipped with the addition (a, b) 7→ max(a, b)and the multiplication (a, b) 7→ a + b. The term “tropical” refers to such algebraic structures, inwhich the addition is idempotent. Many classical notions have tropical analogues. In particular,tropical linear spaces can be defined as modules, by replacing the ring of scalars by the max-plussemiring [6,2]. Such spaces have some features in common with classical convex polyhedra, see [4,5].
In this talk, we shall review the properties of projections on tropical spaces.Since these spaces are naturally ordered, we may define the projection of a vector onto a trop-
ical space to be the greatest vector of this tropical space that the former vector dominates. Such
21
projectors appear to be reminiscent of orthogonal projectors: they minimize a metric -the Hilbert’sprojective metric-, and they can be used to prove separation theorems [2]. However, they turn outto be non-linear.
Hence, a natural question is to characterize those tropical linear spaces which are the images oflinear projectors. We shall consider here specially the case of finitely generated subspaces of the n-fold Cartesian product of the max-plus semiring. Such spaces can be identified to the row or columnspaces of tropical matrices. It is known [3] that the row (or column) space of a tropical matrix Ais the image of a linear projector if and only if A is regular in the sense of von Neumann, meaningthat there exists a matrix X (a generalized inverse) such that A = AXA. This is equivalent to thecondition that the row (or column) space of A is a projective space.
The main result of this talk is the following characterization, which extends a theorem proved byZaretski [7] in the case of Boolean matrices: a tropical matrix is von Neumann regular if and onlyif its row (or column) space is a completely distributive lattice. When A has full column (or row)rank, the latter condition is equivalent to requiring the row (or column) space of A to be stable bythe infimum operation. Moreover, there is always a linear projector on the complete lattice closureof the column (or row) space of A.
We shall finally indicate an application of these ideas. Projectors on tropical spaces have beenused in [1] to design an analogue of the finite element method, adapted to first order Hamilton-Jacobi equations in which the Hamilonian is convex in the adjoint variable. Such equations governthe evolution of the value function of deterministic optimal control problems. In the classical finiteelement method, the solution is approximated by its projection on a space generated by finiteelements. Essentially the same is true in the tropical case.
References1. M. Akian, S. Gaubert, and A. Lakhoua, The max-plus finite element method for solving
deterministic optimal control problems: basic properties and convergence analysis, SIAM J. Controland Opt., arXiv:math.OC/0603619, to appear.
2. G. Cohen, S. Gaubert, and J.-P. Quadrat, Duality and separation theorems in idempo-tent semimodules, Linear Algebra and Appl., 379: 395–422, 2004, arXiv:math.FA/0212294.
3. G. Cohen, S. Gaubert, and J.P. Quadrat, Linear projectors in the max-plus algebra, InProceedings of the IEEE Mediterranean Conference, Cyprus, 1997. IEEE.
4. M. Develin and B. Sturmfels, Tropical convexity, Doc. Math., 9: 1–27, 2004.5. S. Gaubert and R. Katz, Max-plus convex geometry, In R.A. Schmidt, editor, Proceedings
of the 9th International Conference on Relational Methods in Computer Science and 4th Interna-tional Workshop on Applications of Kleene Algebra (RelMiCS/AKA 2006), volume 4136 of LectureNotes in Comput. Sci., pages 192–206. Springer, 2006.
6. G.L. Litvinov, V.P. Maslov, and G.B. Shpiz, Idempotent functional analysis: an alge-braic approach, Math. Notes, 69(5): 696–729, 2001.
7. K.A. Zaretski, Regular elements in the semigroup of binary relations, Uspekhi Mat. Nauk,17(3): 105–108, 1962.
Cyclic projectors in semimodules over the max-plus semiring
Stephane Gaubert∗, Sergeı Sergeev†
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∗INRIA-Rocquencourt, B.P. 105, 78105 Le Chesnay Cedex, France†Department of Physics, Sub-Department of Quantum Statistics and Field Theory, Moscow State
University, Moscow, 119992 Leninskie Gory, Russia∗The research of the first author was supported by the joint RFBR/CNRS grant 05-01-02807.
†The research of the second author was supported by the RFBR grant 05-01-00824 and the jointRFBR/CNRS grant 05-01-02807.
∗[email protected], †[email protected]
We study the properties of cyclic projectors in semimodules over the max-plus or tropical semiringRmax = (R ∪ −∞,⊕ = max,⊙ = +). A cyclic projector is an operator of the form Pk . . . P1,where for i = 1, . . . , k, Pi is the projector onto a closed subsemimodule Vi ⊆ Rn
max, the action ofwhich is defined by Pi(y) = maxx ∈ Vi : x y.
The study of these cyclic projectors is motivated, on the one hand, by the observation, due toCuninghame-Green and Butkovic [3], that they give rise to an effective pseudo-polynomial algorithmfor solving two-sided systems of equations of the form A ⊙ x = B ⊙ y over the max-plus semiring.On the other hand, projectors Pi have been used to establish separation theorems in idempotentconvex geometry, see works by Cohen, Quadrat, Singer, and the first author [1], [2], and by Litvinov,Maslov, and Shpiz [4].
First we show that the orbit of a cyclic projector maximizes a certain objective function. Wecall this function Hilbert’s value of semimodules, as it is a natural extension of Hilbert’s projectivemetric. Then we characterize the (non-linear) spectrum of cyclic projectors in terms of Hilbert’svalues.
The observation that cyclic projectors are closely related to idempotent separation theoremsleads us to a new separation result: that several closed semimodules with a trivial intersection canbe separated from each other. This means that for each of these semimodules, we can construct anidempotent halfspace containing it, in such a way that the intersection of these halfspaces is trivial.As a corollary, we get an idempotent analogue of Helly’s theorem (another proof of which has beenobtained by Meunier and the first author in an independent work).
Some of our results still hold if the max-plus semiring is replaced by a conditionnally complete,or “b-complete” [4], idempotent semifield.
References1. G. Cohen, S. Gaubert, and J.P. Quadrat, Duality and separation theorems in idempotent
semimodules, Linear Algebra Appl., 379: 395–422, 2004. E-print arXiv:math.FA/0212294.2. G. Cohen, S. Gaubert, J.P. Quadrat, and I. Singer, Max-plus convex sets and func-
tions, In G. Litvinov and V. Maslov, editors. Idempotent Mathematics and Mathematical Physics,volume 377 of Contemporary Mathematics , pages 105–129. AMS, Providence, 2005. E-printarXiv:math.FA/0308166.
3. R.A.Cuninghame-Green and P.Butkovic, The equation A ⊗ x = B ⊗ y over (max,+),Theoretical Computer Science, 293:3–12, 2003.
4. G.L. Litvinov, V.P. Maslov, and G.B. Shpiz, Idempotent functional analysis. An alge-braical approach, Math. Notes, 69(5):696–729, 2001. E-print arXiv:math.FA/0009128.
The direct projection method and its modificationfor solving systems of linear equations with sparse unstructured matrices
23
Gogoleva S. Y.The Samara State Aerospace University named after academician S.P. Korolev,
Applied mathematics department, Russia, 443086, Samara, Moscow Shosse, 34. Phone (846)332-56-07
gogoleva [email protected]
The mathematical models of many practical problems result in systems of linear algebraic equa-tions (SLAE) with large and sparse matrices of coefficients. When the large part of matrix coefficientsconsists of zero, it is quite obvious, that we try to store only nonzero elements. The serious problemat a storage and processing of sparse matrices is represented by fill-in, i.e. occurrence of new nonzeroelements. Reduction of fill-in accompany reduction of the requirements to memory volume and workacceleration of method.
In the given work the direct projection method (DPM) of solving SLAE is considered [1]. In thismethod if to carry out analogy to methods based on decomposition of matrices (LU-decomposition),the matrix of equations system A is factored into two matrices LA and R. [1] For realization DPMprobably to store in operative memory only matrix R, that requires accordingly n(n+1)/2 machinewords. Thus, DPM gives a prize in required operative memory volume twice, in comparison withthe computation scheme based on LU - decomposition, that it is favourable to use at the solvingof large sparse SLAE. The total number of arithmetic operations required for DPM is estimated bysize 2n3/3 +O(n2). It almost as much, how many in a method based on LU - decomposition.
At the solving SLAE with sparse matrices by DPM the fill-in occurs only in a matrix R, whereasin LU - decomposition the fill-in can occur, both in a matrix L, and in a matrix U .
In a case for SLAE with the sparse unstructured matrices the numerical researches of DPM werecarried out on examples considered in [2]. The results of numerical researches have shown, thatat the solving of SLAE with the given matrices DPM surpasses LU - decomposition in arithmeticoperations number and operative memory.
To increase numerical stability DPM apply pivot strategy. At work with sparse matrices, usinga choice of a pivot , it is necessary to keep matrix sparsity. If to consider LU decomposition, in thiscase use the Markowitz strategy. The Markowitz strategy has two lacks.
It is necessary to look through all elements active submatrix, we can encounter numerical insta-bility.
The second lack is connected to that circumstance, that, if as pivot the very small elements willbe chosen, in result we shall receive the large errors in a vector of the decisions.
Offered DPM with a pivoting strategy allows to find the compromise between matrix sparsitypreservation and stability.
References1. Zhdanov A.I., A direct consecutive method of the solving systems of linear algebraic equa-
tions, Report RAN, 1997. Vol. 356. No 4. P. 442-444.2. Østerby O., Zlatev Z., Direct methods for sparse matrices, Springer Verlag, 1983. 127pp.
Matrix Computations and the Secular Equation
Gene H. Golub
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Stanford [email protected]
The “secular equation” is a special way of expressing eigenvalue problems in a variety of applica-tions. We describe the secular equation for several problems, viz. eigenvector problems with a linearconstraint on the eigenvector and the solution of eigenvalue problems where the given matrix hasbeen modified by a rank one matrix. Next we show how the secular equation can be approximatedby use of the Lanczos algorithm. Finally, we discuss numerical methods for solving the approximatesecular equation.
Application of Radon transform for fast solution of boundary value problems forelliptic PDE in domains with complicated geometry
Alexander GrebennikovBenemerita Universidad Autonoma de Puebla,
Facultad de ciencias Fısico Matematicas,Av. San Claudio y Rio verde, Ciudad Universitaria, CP 72570, Puebla, Mexico
Simulation of two-phase incompressible flow:discretization and preconditioning
Sven GrossIGPM, RWTH Aachen University, Germany
Two-phase systems play an important role in chemical engineering. Two examples are extractioncolumns where mass transport takes place between bubbles and a surrounding liquid (fluid-fluidsystem), or falling films which are e.g. used for cooling by heat transfer from a thin liquid layer tothe gaseous phase (liquid-gas system).
In this talk we consider 3D flow simulations of such two-phase systems applying a finite elementmethod on adaptive tetrahedral grids. For tracking the interface we use a level set approach. Theeffect of surface tension is modeled by an interfacial force term (CSF model). Numerical results fora single bubble obtained by our software package DROPS will be presented at the end of the talk.
We will focus on several numerical challenges related to the solution of incompressible two-phase flow problems. Special care has to be taken for the numerical treatment of surface tension,as otherwise artificial spurious velocities are induced at the interface. We therefore analyze theLaplace-Beltrami discretization of the interfacial force term and introduce an extended finite elementspace for the representation of the discontinuous pressure (XFEM). The preconditioning of the Schur
25
complement has to be adapted to the jumping coefficients in the PDE, which are induced by differentmaterial properties of the two phases. First results for the preconditioning of the discontinuouspressure Schur complement are given, too.
Primitivity Preservers for Matrix Tuples
Alexander E. GutermanMoscow State University, Russia
This talk is based on the joint work with LeRoy B. Beasley, Utah State University.Let S be a semiring. A matrix A ∈ Mn(S) is primitive if there is an integer k > 0 such that
all entries of Ak are non-zero. The investigation of certain multi-dimensional dynamical systemsleads to the following notion, which is the generalization of primitivity. Let A,B ∈ Mn(S) andh, k be some non-negative integers. The (h, k)-Hurwitz product is the sum of all matrices which areproducts of h copies of A and k copies of B. We denote it by (A,B)(h,k). A pair (A,B) ∈M2
n(S) iscalled primitive if there exist non-negative integers h, k such that the matrix (A,B)(h,k) is positive.We say that an operator,
T :Mn(S)×Mn(S)→Mn(S)×Mn(S),
preserves primitivity for pairs if for primitive pair (A,B) we have that T (A,B) is also primitive.
Theorem. Let B be a binary Boolean algebra,
T :Mn(B) ×Mn(B)→Mn(B) ×Mn(B)
be a surjective additive operator which preserves primitive pairs. Then there is a permutation matrixP and permutations σ and τ of 1, · · · , n such that one of the following holds:
T (X,Y ) = (PDσ(X)P t, PDτ (Y )P t) for all (X,Y ) ∈Mn(B)×Mn(B);T (X,Y ) = (PDτ (Y )P t, PDσ(X)P t) for all (X,Y ) ∈Mn(B)×Mn(B);T (X,Y ) = (PDσ(Xt)P t, PDτ (Y t)P t) for all (X,Y ) ∈Mn(B)×Mn(B);T (X,Y ) = (PDτ (Y t)P t, PDσ(Xt)P t) for all (X,Y ) ∈Mn(B)×Mn(B),
here Dξ : Mn(S) → Mn(S) is a diagonal replacement operator corresponding to a permutationξ ∈ Sn, i.e. there exists ξ ∈ Sn such that Dξ(Ei,j) = Ei,j whenever i 6= j, and Dξ(Ei,i) = Eξ(i),ξ(i)
for all i.
We also provide the generalization of this theorem for k-tuples of matrices. We obtain analogsof these results in the case of arbitrary antinegative semirings without zero divisors, in particular,this includes non-negative matrices and matrices over max-algebras.
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Fast and stable linear algebra
Olga HoltzUniversity of Berkeley, [email protected]
I will discuss my recent joint work with J.Demmel, I. Dumitiru and R.Kleinberg. The first partof the talk will be devoted to the complexity and stability of matrix multiplications, the second toimplications of our results for other problems or numerical linear algebra.
In the first part, we perform forward error analysis for a large class of recursive matrix mul-tiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrixmultiplication, Numer.Math. 36 (1980), 63–72]. As a consequence of our analysis, we show thatthe exponent ω of matrix multiplication (the optimal running time) can be achieved by numeri-cally stable algorithms. We also show that new group-theoretic algorithms proposed in [H. Cohn,and C.Umans, A group-theoretic approach to fast matrix multiplication, FOCS 2003, 438–449]and [H. Cohn, R.Kleinberg, B. Szegedy and C.Umans, Group-theoretic algorithms for matrix mul-tiplication, FOCS 2005, 379–388] are all included in the class of algorithms to which our analysisapplies, and are therefore numerically stable. We perform detailed error analysis for some specificfast group-theoretic algorithms.
In the second part, we extend our results to show that essentially all standard linear algebra op-erations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion,solving least squares problems, (generalized) eigenvalue problems and the singular value decompo-sition can also be done stably (in a normwise sense) in O(nω+η) operations if matrix multiplicationcan be done in O(nω+η) operations for η > 0 arbitrarily small.
Rational iterations for matrix functions
Bruno IannazzoDipartimento di Fisica e Matematica, Universita dell’Insubria, Via Valleggio, 11, 22100 Como,
Rational iterations are commonly encountered in the study and computation of roots of polyno-mial equations. The generalization to the matrix case appears, for instance, in the computation ofmatrix functions and in the numerical solution of certain matrix equations.
It raises problems somehow new: there can be infinite fixed points, the lack of commutativ-ity of the matrix product can have effects on the convergence in finite arithmetic, it is not evenstraightforward how to define a rational matrix iteration.
We discuss these problems, providing general convergence results for a class of rational matrixiterations. Then, we prove some properties of specific classes of rational iterations.
In particular it is shown that the principal Pade family of iterations for the matrix sign functionand the matrix square root is a special case of rational iterations due to Schroder. This characteri-zation provides a family of iterations for the matrix pth root which preserve the structure of groupof automorphisms associated with a scalar product. The first iteration in that family is the Halleymethod for which is proved a convergence result. Finally, it is shown that the Schur-Newton method
27
for the matrix pth root applied to the inverse Newton iteration can be applied also to the direct caseand to the Halley method.
On regularity in the sense of von Neumann of complete matrix semiringsover additively regular semirings
S. N. Ilyin, D. R. TimurshinKazan State University, Russia
A semiring is an algebraic system (S,+, ·, 0) such that (S,+, 0) is a commutative monoid, (S, ·)is a semigroup, (a+ b)c = ac+ bc, a(b+ c) = ab+ac and a·0 = 0·a = 0 for all a, b, c ∈ S. A semiringS is regular (additively regular) iff an equation axa = a (resp., a + x + a = a) is solvable for anya ∈ S. A semiring S is additively idempotent iff a + a = a for all a ∈ S. Clearly, all rings andadditively idempotent semirings are additively regular. A semiring S is regular in the sense of vonNeumann iff for each a ∈ S there exist x, y ∈ S such that a + axa = aya. Obviously, any regularsemiring is regular in the sense of von Neumann. Conversely, any ring is regular if it is regular inthe sense of von Neumann but this is not true for an arbitrary semiring.
Let S be a semiring and Mn(S) be the semiring of all n×n-matrices over S. It is well-known thatif R is a ring then Mn(R) is regular iff R is regular. The regularity criterion for matrix semirings isgiven in [1], in particular, for n ≥ 3 Mn(S) is not regular if S is not a regular ring. On the otherhand, by [2] Mn(S) is regular in the sense of von Neumann for all n if S is a commutative semiringwith identity 1 and for any a, b ∈ S (a 6= b) there exist x, y ∈ S such that 1 + ax + by = ay + bx.Also, it is easy to prove the following result:
Proposition Let S be an additively idempotent semiring. Then Mn(S) is regular in the sense ofvon Neumann iff so is S.
So, the equivalence of regularity in the sense of von Neumann for S and Mn(S) holds for any ringand additively idempotent semiring. The theorem below gives an analogous result for additivelyregular semirings which satisfy some additional conditions.
We denote by I(S) and R(S) the ideal of all additively idempotent elements in S and the idealof all elements in S which have additive inverses, respectively.
Theorem Let S be an additively regular semiring which is regular in the sense of von Neumann. Ifone of the following conditions holds
1) there exists u ∈ I(S) such that x+ u = u for all x ∈ I(S),2) R(S) or I(S) are finite,
then for any n Mn(S) is regular in the sense of von Neumann.
References1. S.N.Ilyin, Regularity criterion for complete matrix semirings, Mat. Zamet. 70 (2001),
366–374.2. Shamik Ghosh, A note on regularity in matrix semirings, Kyungpook Math. J. 44 (2004),
1–4.
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On the conjugate and semi-conjugate direction methodswith preconditioning projectors
V.P.Il’inInstitute of Computational Mathematics and Mathematical Geophysics, SBRAS, Novosibirsk
The goal of presented work consists in the improvement of an iterative convergence rate andcorresponding estimates for multiplicative and additive type algorithms with preconditioning pro-jectors, applied to domain decomposition approach and investigated by J.Bramble with his coauthors(1991). For the multiplicative method, the symmetrized alternating direction algorithm version isproposed. It can be accelerated by means of the conjugate gradient or conjugate residual algorithmsin Krylov subspaces. The similar acceleration is suggested for the additive type iterative procedure.The acceleration of the multiplicative method without symmetrization is done by semi-conjugateresidual algorithm which is stable modification of GCG method by H.Elman and others (1983).
It is shown that the multiplicative and additive projector algorithms include, as the particularcases, the known Kaczmarz’s (1937) and Cimmino’s (1938) methods respectively. Various general-izations and acceleration in Krylov subspaces of these algorithms are proposed. The estimations ofthe iterative convergence rate are obtained. The results of the numerical experiments for the modelgrid diffusion-convection boundary value problems, with different coefficients and mesh steps, arepresented.
On the quaternionic equation AX +BXC +XD = E
Drahoslava Janovska∗, Gerhard Opfer†∗Institute of Chemical Technology, Department of Mathematics, Prague, Czech Republic
†University Hamburg∗[email protected]
We study the quaternionic equation AX +BXC+XD = E. For BC = 0 the resulting equationis called Sylvester’s equation. For this case a complete solution will be given. For the general casewe show that the solution can be found by solving a corresponding matrix equation.
On singular values of parameter dependent matrices
Drahoslava Janovska∗, Vladimır Janovsky†
∗Institute of Chemical Technology, Department of Mathematics, Prague, Czech Republic†Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
The research of both authors was partially supported by the Grant Agency of the Czech Republic(grant No. 201/06/0356).
∗[email protected], †[email protected]
29
We consider the Analytic Singular Value Decomposition, ASVD, of matrix valued functions.ASVD is smooth up to isolated parameter values at which either a multiple singular value or azero singular value turns up on the path. These exceptional points are called non-generic, see [1].They were classified in [2]. Note that ASVD-computations, see e.g. [1,2], require information on allsingular values on the path and hence the algorithms were not able to cope with large sparse inputdata.
In [3], we investigated a pathfollowing of just one simple singular value and the correspondingleft/right singular vector. A breakdown of the continuation is related to non-generic points on thepath. We apply Singularity Theory to analyze and classify these non-generic points. Our analysiswill include the questions concerning structural stability. The classification will result in preciselocalization technique of these points. We compare our classification list with [2].
References1. A. Bunse-Gerstner, R. Byers, V. Mehrmann, and N. K. Nichols, Numerical Com-
putation of an Analytic Singular Value Decomposition of a Matrix Valued Function, Numer. Math.,60 (1991), pp. 1–39.
2. K. Wright, Differential equations for the analytic singular value decomposition of a matrix,Numer. Math. 63 (1992), pp. 283–295.
3. D. Janovska, V. Janovsky, The analytic SVD: On the non-generic points on the path, Pro-ceedings of Joint GAMM-SIAM Conference on Applied Linear Algebra, Dusseldorf 2006. Submittedto ETNA.
Multilevel Optimal Preconditioners
Xiao-Qing JinDepartment of Mathematics, University of Macau, Macao, China
Incomplete LU factorization of general sparse matrices
Igor KaporinComputing Center of Russian Academy of Sciences,
Vavilova 40, Moscow 119991, [email protected]
We discuss a convergence estimate for Preconditioned GMRES-type iterative linear equationsolvers. Formulation of the result is specifically adjusted to the case when the preconditioning isbased on an Incomplete Triangular factorization “by value”.
30
Consider the linear algebraic system Ax = b with general unsymmetric nonsingular sparse n×nmatrix A. The ILU preconditioned GMRES-type iterative methods use the preconditioner matrixC ≈ A of the form C = PLLUPR as is obtained from the ILU equation
A = PLLUPR +E,
where L and U are nonsingular lower and upper triangular matrices, respectively, while PL and PR
are permutation matrices. The additive term E is the ILU error matrix, for which the magnitudeof its entries are limited by a prescribed threshold value τ . Under a proper choice of permutationmatrices PL and PR, one can also observe that not only the stability of the triangular factors isimproved (indeed, it seems natural to require at least that cond(C) ≤ cond(A)), but also ‖E‖2F ≡trace(ETE) = O(nτ2). The latter means that only relatively few nonzero entries of E attain theupper bound τ by their magnitude.
Furthermore, as was noted by many authors, see, e.g. [2,4] and references therein, the ILUfactorization ‘by value’ applied to a properly two-side scaled coefficient matrix AS = DLADR mayyield a much better preconditioning, especially in several hard-to-solve cases, see also [8].
Main Result. Based on the latter paper, we present the following superlinear residual normbound for GMRES iterations:
‖rk‖‖r0‖
≤ κ K(AS)(4en
ksin2[CS , AS ]
)k/2, e = exp(1), κ = ‖V ‖‖V −1‖, (1)
whereK(B) =
(n−1/2‖B‖F
)n /|detB| (2)
is the unsymmetric K-condition number of a nonsingular matrix B, rk = b−Axk is the kth residualgenerated by Preconditioned GMRES(∞), CS is the preconditioner for AS and V is the n×n matrixof eigenvectors of AC−1 (i.e., we assume the diagonalizability of the preconditioned matrix).
Note that the earlier estimate of similar type developed in [7] was formulated in terms ofsin[In, AC
−1] and λmin(AC−1), which quantities are very problematic to measure even a posteri-
ori.The correctness of the above convergence estimate was also tested numerically using many “hard”
test matrices taken from the University of Florida Sparse Matrix Collection [1].Relation to Scaling. In view of (1), it is natural to require that the scaling should minimize
the functional (2) with B = AS . It appears that the minimizer satisfies exactly the requirement thatAS = DLADR has Euclidean norms of each row and column equal to the same number, exactly aswas recommended in [2,4].
Relation to ILU Error. Let the matrix be scaled in such a way that ‖AS‖2F = n. Then
sin2[CS , AS ] ≡ 1−(trace(AT
SCS))2
‖CS‖2F ‖AS‖2F=
minσ ‖AS − σCS‖2F‖AS‖2F
≤ ‖AS −CS‖2F ‖AS‖−2F = ‖ES‖2F ‖AS‖−2
F = n−1‖ES‖2F .
Hence, the above convergence estimate takes the form
‖rk‖‖r0‖ ≤
κ
|detAS |
(3.3√k‖ES‖F
)k/2
,
cf. [8]. It remains to note that if the threshold parameter is chosen sufficiently small, e.g. τ = 0.001,and the ILU factors are stable enough, then the typical values of ‖ES‖F are not big (often one canobserve ‖ES‖F < 1 even for rather big problems).
31
Conclusions. First, a theoretical justification is found for the standard pre-scaling techniquerelated to the ILU factorization, with implications for the practical implementation. (Namely, amore accurate evaluation of DL and DR may be useful, and faster algorithms should be developedfor this purpose.) Second, an even better estimate for the reduction of the original (unscaled) residialcan be obtained in terms of the scaled ILU error.
References1. University of Florida Sparse Matrix Collection. http://www.cise.ufl.edu/research/sparse/
matrices/2. V.F. de Almeida, A.M. Chapman, and J.J. Derby, On Equilibration and Sparse Factor-
ization of Matrices Arising in Finite Element Solutions of Partial Differential Equations, Numer.Methods Partial Different. Equ. 16, 11–29, 2000.
3. M. Bollhofer and Y. Saad, Multilevel preconditioners constructed from inverse-basedILUs, SIAM J. Sci. Comput., 27, 1627–1650, 2006.
4. O. E. Livne and G. H. Golub, Scaling by Binormalization, Numer. Alg., 35, 97–120, 2004.5. M. H. Schneider and S. A. Zenios, A comparative study of algorithms for matrix balancing,
Operations Research, 38, 439–455, 1990.6. N. Li, Y. Saad, and E. Chow, Crout versions of ILU for general sparse matrices, SIAM
J. Sci. Comput., 25, 716–728, 2003.7. I. Kaporin, Superlinear convergence in minimum residual iterations, Numerical Linear Al-
gebra with Applications 12, pp. 453–470, 2005.8. I. Kaporin, Scaling, Reordering, and Diagonal Pivoting in ILU Preconditionings, (29 pp.)
To appear in Russian Journal of Numerical Analysis and Mathematical Modelling 12, no.4, 2007.
Recursive scaling, permutation and2×2-block splitting in ILU preconditionings
Igor Kaporin and Igor KonshinComputing Center of Russian Academy of Sciences,
Vavilova 40, Moscow 119991, [email protected]
We discuss a technique for the Incomplete LU preconditioning of general unsymmetric nonsin-gular matrix aimed at the construction of robust linear equation solvers. The idea of such precon-ditioning (multilevel ILU) was proposed in [3]. The key component of the technique is a recursiveconstruction of a 2× 2 block splitting with diagonally dominant leading block.
In the present work, we propose a new algorithm for the construction of such splittings.Consider the linear algebraic system Ax = b with a nonsingular general sparse n × n matrix
A. In the construction of preconditioned GMRES-type iterative methods, the key problem is toconstruct the preconditioner C ≈ A. By the substitution x = C−1y the original problem is thenreduced to the preconditioned system My = b where M = AC−1 is the right preconditioned matrix.An iterative scheme is then applied to the latter problem. This normally yields a faster convergenceof the iterations.
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Let n1 and n2 be the block sizes of the splitting, n1 + n2 = n. The preconditioner C is obtainedby making the first n1 steps of an ILU factorization ‘by value’ of the Crout type [6]:
CSR = ASR − E =
[A
(SR)11 A
(SR)12
A(SR)21 A
(SR)22
]− E
=
[L11 0L21 I2
] [I1 0
0 A
] [U11 U12
0 I2
],
(1)
whereE is the ILU error matrix (with the magnitudes of its entries limited by a prescribed threshold),the n2 × n2-matrix
A = A(SR)22 − L21U12
is the approximate ‘active submatrix’ (or Schur complement) and
ASR = PLASPR, AS = DLADR
are the scaled-and-reordered and two-side scaled coefficient matrices, respectively. Here PL, PR arepermutation matrices and DL, DR are nonsingular diagonal matrices. The preconditioner C for theoriginal matrix A can be found from the relation CSR = PLDLCDRPR.
If the size of A in this decomposition is still large, then this construction is applied recursively,i.e., A is replaced by C in the same way. When the size of A is reduced to an acceptable level, itcan be processed using the ‘direct’ sparse LU factorization.
As is shown in [3], in order to guarantee a good preconditioning quality in (1) one should imposethe following two contradictive requirements: (i) the size n1 of the leading block in splitting (1)is as large as possible and (ii) the same leading block in (1) is well-conditioned. Typically, A11 isconstructed as a diagonally dominant matrix, cf. [3]. Note that in our treatment we consider thediagonal dominance with respect to the scaled matrix AS .
Two-side scaling stage. Similar to [2,4], we determine the scaling by (nearly) equalizingboth row and column 1-norms of the scaled matrix AS = DLADR, i.e. we approximately solvethe nonlinear system
∑j |(AS)ij | = 1,
∑i |(AS)ij | = 1. The diagonal matrices DL and DR were
obtained by the RAS (Row-Alternating Scaling) iterations (see, e.g. [5] and references therein).Pre-ordering stage. As the algorithm for two-side unsymmetric reordering of the matrix AS ,
we used the simplest possible ‘greedy’ strategy, which constructs the pre-ordered matrix ASR =
PLASPR such that |(ASR)kk| ≥ |(ASR)ij |, i > k, j > k. If the array of nonzero elements of H isorganized as binary heap, the corresponding algorithm requires O(nz(A) log nz(A)) operations.
Splitting stage. Let us denote the entries of the scaled and pre-ordered matrix as aij = (ASR)ij .We sequentially construct the set D of n1 diagonal positions (to be included in the required leadingblock) by checking the diagonal dominance condition:D = ∅; for k = 1, 2, ..., n
if(∑
j∈D∪k\i |ai,j | < |ai,i|, 1 ≤ i ≤ k)
then D := D ∪ kend for
Using a data structure which provides a fast access both to ASR and ATSR
, one can implement theabove procedure in O(nz(A)) operations.
In order to gather consecutively the positions defined by the index set D, one applies the corre-sponding symmetrical reordering to ASR thus obtaining the final 2× 2 split form: ASR = P TASRP ,
i.e., PL = P T PL, PR = PRP .Note that the value n1 = |D| determined as above, typically grows with the number of scaling
iterations used. In practice, we performed the RAS iterations until the relative spread of the row orcolumn norms be reduced below 1 + δ, with δ in the range [0.1, 0.5].
33
The efficiency of the proposed procedure was tested numerically using many ‘hard’ test matricestaken from the University of Florida Sparse Matrix Collection [1].
References1. University of Florida Sparse Matrix Collection. http://www.cise.ufl.edu/research/sparse/
matrices/2. V.F. de Almeida, A.M. Chapman, and J.J. Derby, On Equilibration and Sparse Factor-
ization of Matrices Arising in Finite Element Solutions of Partial Differential Equations, Numer.Methods Partial Different. Equ. 16, 11–29, 2000.
3. M. Bollhofer and Y. Saad, Multilevel preconditioners constructed from inverse-basedILUs, SIAM J. Sci. Comput., 27, 1627–1650, 2006.
4. O. E. Livne and G. H. Golub, Scaling by Binormalization, Numer. Alg., 35, 97–120, 2004.5. M. H. Schneider and S. A. Zenios, A comparative study of algorithms for matrix balancing,
Operations Research, 38, 439–455, 1990.6. N. Li, Y. Saad, and E. Chow, Crout versions of ILU for general sparse matrices, SIAM
J. Sci. Comput., 25, 716–728, 2003.
Monotone matrices and finite volume schemes for diffusion problems preservingnon-negativity of solution
I. KapyrinInstitute of Numerical Mathematics of RAS
A new finite volume scheme for 3D diffusion problems with heterogeneous full diffusion tensor isconsidered. The discretization uses nonlinear two-point flux approximation on unstructured tetra-hedral grids. Monotonicity of the linearized operator allows us to guarantee non-negativity of thediscrete solution.
Spectral model order reduction for control systemsmodelling passive integration circuits
Yu. M. Nechepurenko∗, A. S. Potyagalova†, I. A. Karaseva‡
∗Institute of Numerical Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow,119333 Russia,
†Cadence Design Systems LLC, ul. B.Ordynka 44, Bldg.4, Moscow, 119017 Russia,‡Moscow Institute of Physics and Technology, ul. Kerchenskaya 1a, Bldg.1, Moscow, 117303.
This work was supported by the Russian Foundation for Basic Research (project 07-01-00658)and Russian Academy of Sciences (project “Optimization of numerical algorithms for solving the
problems of mathematical physics”)∗[email protected], †[email protected], ‡irina−[email protected]
34
Complexity reduction for generating compact models of interconnect RCLM networks have beenan intensive research area in the past decade due to increasing signal integrity effects and risingcouplings modelled with parasitic capacitors and inductors. Since the original systems were passiveone of the main requirements to the reduced systems was the preservation of passivity. Many alge-braic model order reduction methods preserving passivity were proposed based on implicit momentmatching, congruence transformations and truncated balance realizations.
In this report we present a novel algebraic spectral model order reduction algorithm equippedwith efficient tools for preserving the passivity. For RC networks our approach is similar to the well-known spectral reduction technique PACT (pole analysis via congruence transformations) and canbe construed as its generalization. Up to the present any such a generalization preserving passivityhas seemed impossible and other model order reduction methods were applied for the reduction ofRCL and RCLM networks.
The accuracy and reduction ratio of resulting reduced-order models are demonstrated with severalindustrial examples.
Toeplitz and Toeplitz-block-Toeplitz matrices and their correlation with syzygies ofpolynomials
Houssam Khalil∗, Bernard Mourrain†, Michelle Schatzman∗
∗Institut Camille Jordan, Universite Claude Bernard Lyon143 boulevard du 11 novembre 191869622 Villeurbanne cedex France
†INRIA, GALAAD team2004 route des Lucioles
BP 93, 06902 Sophia Antipolis Cedex, [email protected], [email protected], [email protected]
Structured matrices appear in various domains, such as scientific computing, signal process-ing, . . . They usually express, in a linearized way, a problem which depends on less parameters thanthe number of entries of the corresponding matrix. An important area of research is devoted to thedevelopment of methods for the treatment of such matrices, which depend on the actual parametersinvolved in these matrices.
Among well-known structured matrices, Toeplitz and Hankel structures have been intensivelystudied. Nearly optimal algorithms are known for the multiplication or the resolution of linearsystems, for such structure. Namely, if A is a such matrix of size n, multiplying it by a vector orsolving a linear system with A requires O(n) arithmetic operations (where O(n) = O(n logc(n))for some c > 0) arithmetic operations for some c > 0. Such algorithms are called super-fast, inopposition with fast algorithms requiring O(n2) arithmetic operations. The correlation with othertypes of structured matrices has also been well developed in the literature, allowing to treat soefficiently other structures such as Vandermonde or Cauchy-like structures.
Such problems are strongly connected to polynomial problems. For instance, the product of aToeplitz matrix by a vector can be deduced from the product of two univariate polynomials, andthus can be computed efficiently by evaluation-interpolation techniques, based on FFT. The inverseof a Hankel or Toeplitz matrix is connected to the Bezoutian of the polynomials associated to theirgenerators.
35
However, most of these methods involve univariate polynomials. So far, few investigations havebeen pursued for the treatment of multilevel structured matrices, related to multivariate problems.Such linear systems appear for instance in resultant or in residue constructions, in normal formcomputations, or more generally in multivariate polynomial algebra. A main challenge here is todevise super-fast algorithms of complexity O(n) for the resolution of multi-structured systems ofsize n.
In this paper, we consider block-Toeplitz matrices, where each block is a Toeplitz matrix. Sucha structure, which is the first step to multi-level structures, is involved in many bivariate problems,or in numerical linear problems. We re-investigate first the resolution of Toeplitz systems T u = g,from a new point of view, by correlating the solution of such problems with syzygies of polynomialsor moving lines. We show an explicit connection between the generators of a Toeplitz matrix andthe generators of the corresponding module of syzygies. We show that this module is generated bytwo elements of degree n and the solution of T u = g can be reinterpreted as the remainder of anexplicit vector depending on g, by these two generators.
This approach extends naturally to multivariate problems and we describe for Toeplitz-block-Toeplitz matrices, the structure of the corresponding generators. In particular, we show the knownresult that the module of syzygies of k non-zero bivariate polynomials is free of rank k− 1, by a newelementary proof.
Exploiting the properties of moving lines associated to Toeplitz matrices, we give a new waywhich, may be, we can use it to compute explicitly the solution of a Toeplitz-block-Toeplitz matriceswith the complexity of such computation is bounded by O(mn) where n is the number of Toeplitzblocks and m their size.
Sparse Approximation of FEM Matrix for Sheet Current Integro-DifferentialEquation
M. M. Khapaev∗, M. Yu. KupiyanovComp. Sci. Dept. of Moscow State University, Russia
In the report we consider the problem of effective numerical solution of 2D sheet current integro-differential equations. Typically the equations for sheet current are derived from Maxwell equationsfor magnetic potential. The equation contain weak singularity in the kernel and for normal conduc-tors has the form
Rs~J(r) +
jω
4π
∫ ∫
S
~J(r′)
|r − r′|ds = −∇Φ(r), ∆Φ(r) = 0, r ∈ S. (1)
Here S is 2D domain, r = (x, y). ~J(r), Φ(r) are phasors for sheet current and voltage. ω is frequency,Re(Rs) ≥ 0, Im(Rs) ≥ 0 — thin plate resistance [1]. For (1), well developed numerical techniquesexist [1]. In many cases, the numerical solution of (1) is very time consuming even if GMRES andFast Multipoles Method for kernel approximation are used [1].
We are interested in the solution of very similar to (1) task for superconductors [3]. In our caseequations are real and does not depend on frequency. The equations we solve are singular and follows
36
from Biot-Savart formula
λ⊥
(∇× ~J(r)
)z+
1
4π
∫∫
S
(~J(r)× (∇r ·
1
|r − r0|)
)
z
dxdy = 0, (2)
where r = (x, y), r0 = (x0, y0), r, r0 ∈ S, λ⊥ is real coefficient. Equation (2) must be fulfilled bythe boundary conditions for ~J(r). On the part of the boundary of S boundary conditions reducesto Jn = 0, on the rest of the boundary the non-zero value of Jn is specified. Equation (1) can bereduced to form similar to (2) using 2D rotor differentiation.
To solve (2) we first rewrite it using stream function ψ(r). Then we obtain the problem verysimilar to first boundary problem for Laplace equation:
−λ⊥∆ψ(r0) +1
4π
∫∫
S
(∇ψ(r),∇r
1
|r − r0|
)ds = 0, ψ(r) = F (r), r ∈ ∂S. (3)
The function F (r) is completely defined by boundary condition for current. Equation (3) for ψ(r)is hypersingular with kernel 1/|r − r0|3. It is known that operator in (3) is positive and self-adjoint[2] (if ψ(r) = 0 on ∂S).
To solve (3) numerically we use finite element method with linear elements on triangular grids[3]. The resulting FEM matrix is dense die to integral operator. Let uh
i , uhj be FEM basis functions,
then matrix elements are
a(uhi , u
hj ) = λ⊥
∫∫
Si∩Sj
(∇uhi ,∇uh
j )ds +1
4π
∫∫
Si
ds
∫∫
Sj
(∇uhj (r′),∇uh
i (r))
|r − r′| ds′, (4)
The method of calculation of quadruple integrals is given in [3].Due to integral term in (3) FEM results in fully populated symmetric matrix. The dimension of
this matrix for practical problems is large (103− 104). This fact is the key limitation of the method.We suggest a simple method for sparsifying the FEM matrix. We simply drop small elements.
Symmetry of matrix is preserved and positive definetness also can be preserved if dropped elementsare small enough.
Consider matrix element for integral operator (4). Diagonal matrrix elements are large andpositive but matrix is not diagonally dominant. Sub-diagonal elements can be of any sign. IfSi
⋂Sj = ∅ then elements are negative and
bij =
∫∫
Si
ds
∫∫
Sj
(∇uhj (r′),∇uh
i (r))
|r − r′| ds′ = −∫∫
Si
ds
∫∫
Sj
uhj (r′)uh
i (r)
|r − r′|3 ds′ (5)
Let h be the estimation of diameters of triangles in the mesh. Then mijh ≈ |ri − rj|, mij > 1 andfrom (5) follows
|bij | ≤Mh
m3ij
. (6)
Thus matrix elements quickly decay if distance between finite elements grows. For equation withweak singularity this decaying is of first order only.
In the report, we present set of results of numerical experiments with sparse FEM matrix ap-proximation and direct linear system solver. The conclusion is that the simple method can solvelarge problems and overcome memory and CPU time limitations of dense matrix technique.
The paper is supported by ISTC project 3174.
References
37
1. M. Kamon, M. J. Tsuk and J. K. White, FastHenry: a multipole-accelerated 3-D in-ductance extraction program, IEEE Trans. Microwave Theory and Techn. vol. 42, pp. 1750–1758,September 1994.
2. V. J. Ervin, E. P. Stephan, A Boundary Element Galerkin Method for a HypersingularIntegral Equation on Open Surfaces, Mathematical Methods in the Applied Sciences, 1990, v. 13,issue 4, pp. 281–289.
3. M. Khapaev, Inductance Extraction of Multilayer Finite Thickness Superconductor Circuits,IEEE Trans. Microwave Theory and Techn., 2001, v.49, pp. 217–220.
Some approaches to solving inverse eigenvalue problems
V. N. Kublanovskaya∗, V. B. Khazanov†
∗St.-Petersburg Branch of the V. A. Steklov Mathematical Institute RAS†St.-Petersburg State Marine Technical University
The work was supported by the RFBR (project No. 05-01-00945a).∗[email protected], †[email protected]
The methods are proposed for solution of inverse eigenvalue problems for regular polynomialmatrices (both the one-parameter ones, including the pencils A− λI and A− λB, and the multipa-rameter ones) which are formulated in two ways:
• to replace known eigenvalues of the polynomial matrix by given numbers with the preservationof the right (or left) eigenvectors associated with the invariable eigenvalues,• to calculate values of parameters of a parameter matrix for which the matrix has the prescribed
spectrum points (eigenvalues).
For the first statement the polynomial matrix F (λ) and its eigenvalues λ∗i ∈ σ[F ], i = 1, . . . , p
are given. It is required to compute the polynomial matrix F (λ) with spectrum σ[F ] = σ[F ] \λ∗i , i = 1, . . . , p ∪ µ∗i , i = 1, . . . , p. The right (or left) eigenvectors associated with the invariableeigenvalues must be preserve. The following approaches to solution of the problem are proposed.
1. The transformations of the matrix, based on the rank factorizations of polynomial matrix byusing of unimodular matrices [4]. The factorization F (λ) = F1(λ)F2(λ) is fulfilled, where σ[F1] =λ∗i , i = 1, . . . , p. For each λ∗i the respective eigenspace of the matrix F1(λ) is computed. Afterwardsλ∗i of the corresponding multiplicity is replaced by µ∗i . The procedure is repeated if the geometricmultiplicity λ∗i is smaller then the geometric multiplicity.
2. The transformations of the matrix, based on the generalizations of the exhaustation processes(Vilandt and Hotelling methods) of an eigenvalue of a constant matrix [3]. For each λ∗i the right (orleft) eigenvectors of the matrix F (λ) are computed and the transformations of the matrix are fulfilleduntil all λ∗i are replaced by µ∗i . If the block spectral characteristics are given the transformation isfulfilled the one step.
For the second statement the multiparameter polynomial n × n matrix F (λ,µ) is given andit is required to compute the values of parameters λ = (λ1, . . . , λn) for which the matrix has aprescribed set of the spectrum points µ∗
1, . . . ,µ∗n. In the case of the one-parameter matrix F (λ, µ) –
the prescribed set of the eigenvalues µ∗1, . . . , µ∗n. The following approaches to solution of this problem
are proposed.
38
1. Reduction to the system of nonlinear algebraic equations. By used of the Faddeev tracemethod the polynomial ϕ(λ,µ) := detF (λ,µ) is computed. Once the polynomials ϕi(λ) := ϕ(λ,µ∗
i )are computed, the initial problems is reduced to searching a solution of the system of equationsϕi(λ) = 0, i = 1, . . . , n. The corresponding methods are discussed in [1,2].
2. Reduction to the system of nonlinear equations. The equivalent nonlinear system is consideredinstead of the system of determinant equations. The following characteristics of singularity of matrixcan be used: the equality of the least singular value or the least diagonal elements of the left triangularmatrix obtained by use of the normalized decomposition of constant matrix to zero. The methodbased on this approach is considered, for example, in [5].
3. Reduction to the multiparameter spectral problem. The normalized eigenvectors xi associatedwith the spectrum points µ∗
i , i = 1, . . . , n are introduced and the initial problem is re-formulated asa multiparameter spectral one:
Fi(λ)xi := F (λ,µ∗i )xi = 0 , xT
i xi = 1 , i = 1, . . . , n
In particular, the additive and multiplicative inverse problems are reduced to the problems for thematrix pencils Fi(λ) = Fi−
∑nj=1 λBij. The obtained nonlinear system (which is of the order (n+1)n)
can be solved using iterative methods (i.e. Newton, Tchebychev, Halley, inverse iterations, gradientmethods) [6]. On the each step of these methods (with the exception of the gradient methods) thelinear systems of the order n are solved. The linearization of polynomial matrix F (λ) [7] of thedegree s > 1 is realized by means to transfer to the accompanied pencil. Moreover the order isincreased at sn or
(s+nn
)times.
References1. Kublanovskaya V.N. Solution of systems of nonlinear algebraic eguations. Methods and
algorithms IV, Zap. nauchn. semin. POMI, 248 (1998), 124–146.2. Kublanovskaya V.N. An approach to solving inverse eigenvalue problems for matrices,
Zap. nauchn. semin. POMI, 268 (2000), 950-114.3. Kublanovskaya V.N., Khazanov V.B. Exhaustion in the spectral problems for matrix
pencils, In: Computing processes and systems, 5, (ed. G. I.Marchuk), Moscow, Nauka, 1987,pp. 138–147.
4. Kublanovskaya V.N., Khazanov V.B. Numerical methods for solving parametric prob-lems of algebra. Part 1. One-parameter problems, St. Petersburg, Nauka, 2004.
5. Khazanov V.B. Methods for solving two-parameter eigenvalue problem, Proc. of LKI: Themathematical software for automatized shipbuilding systems, Leningrad, LKI, 1987, pp. 118–125.
6. Khazanov V.B. Methods for solving spectral problems for multiparameter matrix pencils,Zap. nauchn. semin. POMI, 296 (2003), 139–168.
7. Khazanov V.B. To solving spectral problems for multiparameter polynomial matrices, Zap.nauchn. semin. POMI, 334 (2006), 212–231.
On solving spectral problems for nonlinear multiparameter matrices
V. B. KhazanovSt.-Petersburg State Marine Technical University
39
The work was supported by the RFBR (project No. 05-01-00945a)[email protected]
The approach proposed by V. N. Kublanovskaya to solving nonlinear one-parameter spectralproblem (based on normalized decomposition of constant matrix [1]), is generalized for the caseof nonlinear multiparameter matrices.
The joint spectrum of the multiparameter regular matrices Fi(λ) of the order ni, , i = 1, . . . , q,whose elements are the nonlinear differential functions of the parameters λ = (λ1, . . . , λq) is deter-mined as zeros of the system of nonlinear equations
detFi(λ) = 0, i = 1, . . . , q.
The following equivalent equations are used instead of determinant ones:
fi(λ) = l(i)ni(λ), i = 1, . . . , q,
where l(i)ni (λ) is the last diagonal element of the left triangular matrix Li(λ), obtained using the
normalized decomposition of constant matrix Fi(λ) with the fixed value of multiparameter λ:
Θi(λ)Fi(λ) = Li(λ)QTi (λ), i = 1, . . . , q.
Here Θi(λ), Qi(λ) and Li(λ) are the permutation, orthogonal and left triangular matrices of theorder of ni correspondingly.
The assumption that the value of the multiparameter λ belongs to a neighborhood of the spec-trum point λ∗ ∈ σF1, . . . , Fq, where the permutation matrix Θi(λ) is stabilized, allows one toobtain the relationships for computing the partial derivatives of the left hand sides of the consid-ered equations. In its turn it enables one to apply such methods as the Newton method and themethod of steepest descent. The modifications of the methods are considered which avoid solutionof ill-conditioned systems of linear equations.
References1. Kublanovskaya V.N., On applying the Newton method to computing eigenvalues of λ-
matrices, Dokl. AS USSR, 188(5) (1968), 1004–1005.2. Khazanov V.B., Methods for solving two-parameter eigenvalue problem, Proc. of LKI: The
mathematical software for automatized shipbuilding systems, Leningrad, LKI, 1987, pp. 118–125.3. Yakovlev M. I. On some methods for solving nonlinear equations, Proc. MI AS USSR, 84
(1965), 8–40.
Multi-linear approximation of collocatedconvolution transform in Rd
Boris N. KhoromskijMax-Planck-Institute for Mathematics in the Sciences,
Inselstr. 22-26, D-04103 Leipzig, [email protected]
40
Modern methods of tensor-product decomposition allow an efficient data-sparse approximationto integral and more general nonlocal operators in higher dimensions (cf. [1–8]). Examples of suchnonlocal mappings are classical Green’s functions, solution operators of elliptic/parabolic BVPs, in-tegral operators arising from the Hartree-Fock equation in electronic structure calculations, collisionintegrals from the deterministic Boltzmann equation as well as the convolution integrals from theOrnstein-Zernike equation in molecular dynamics.
We discuss the approximation theory and numerical methods of structured low tensor-rank ap-proximation to convolution transform in Rd. In some cases, the asymptotic complexity of suchapproximations can be estimated by O(dn log n+ logq n), where N = nd is the discrete problem size(hence, avoiding the “curse of dimensionality”). In particular, we focus on applications arising fromthe Hartree-Fock equation, which includes the convolution with the classical Newton potential 1
|x−y| ,
x, y ∈ R3.
References1. S.R. Chinnamsetty, H.-J. Flad, W. Hackbusch, V. Khoromskaia and B.N. Kho-
romskij, Tensor approximation in fully discrete electronic structure calculations, MPI MIS, Leipzig2007 (in progress).
2. I. P. Gavrilyuk, W. Hackbusch and B. N. Khoromskij, Tensor-Product Approxi-mation to Elliptic and Parabolic Solution Operators in Higher Dimensions. Computing 74 (2005),131-157.
3. W. Hackbusch, B.N. Khoromskij, and E. Tyrtyshnikov, Hierarchical Kronecker tensor-product approximation, J. Numer. Math. 13 (2005), 119-156.
4. W. Hackbusch and B.N. Khoromskij, Low-rank Kronecker product approximation tomulti-dimensional nonlocal operators. Parts I/II. Computing 76 (2006), 177-202/203-225.
5. B.N. Khoromskij, Structured data-sparse approximation to high order tensors arising fromthe deterministic Boltzmann equation, Preprint 4, MPI MIS, Leipzig 2005 (Math. Comp., to appear).
6. B.N. Khoromskij, Structured Rank-(r1, ..., rd) Decomposition of Function-related Tensorsin Rd, Comp. Meth. in Applied Math., 6 (2006), 2, 194–220.
7. B.N. Khoromskij and V. Khoromskaia, On the Best Rank-(r1, ..., rd) Tucker Approxi-mation to the Classical Potentials, Preprint 105, MPI MIS, Leipzig 2006 (CEJoM 2007, to appear).
8. E. Tyrtyshnikov, Tensor approximation of matrices generated by asymptotically smoothfunctions, Mat. Sb. 194 (6), 2003, 147–160.
Stabilizing and destabilizing effect of breaking the Hamiltonian and reversiblesymmetry
Oleg N. KirillovMoscow State Lomonosov University, Institute of Mechanics, Michurinskii pr. 1, 119192 Moscow,
The eigenvalue problem is studied for the matrix polynomial L(λ) = Iλ2 +(δD+γG)λ+P+νNdetermining the stability of a linear autonomous non-conservative system in presence of dissipative,gyroscopic, potential, and non-conservative positional forces. It is assumed that the matrices in-volved are real, I is the unit matrix, D and P are symmetric, and G and N are skew-symmetric. In
41
practically important cases of potential (δ = γ = ν = 0), gyroscopic (δ = ν = 0), and circulatorysystem (δ = γ = 0) possessing the Hamiltonian or reversible symmetry, the necessary and sufficientcondition for marginal stability is that the eigenvalues λ are purely imaginary and semi-simple. Inmodern problems of wave propagation, acoustics of friction, and fluid-structure interaction it is im-portant to know how the marginal stability is destroyed or improved up to asymptotic stability dueto action of non-Hamiltonian and non-reversible perturbations [1,2,3,4,5,6]. The present contribu-tion shows that in both cases the boundary of the asymptotic stability domain of the perturbedsystem possesses numerous singularities such as “Whitney umbrella” and the “Break of an edge”that govern stabilization and destabilization. With the use of the perturbation theory of multipleeigenvalues approximations of the stability boundary near the singularities and estimates of thecritical gyroscopic and circulatory parameters are found in analytic form. The role of dissipativeforces with the indefinite matrix D is investigated and new explicit condition is obtained for suchforces to be stabilizing. The typical motion of eigenvalues is described. Applications to the stabilityproblems of gyroscopic systems with stationary and rotating damping, such as the Tippe Top andJellet’s egg, as well as to the problem of suppression of the squeal of the automotive disc brake arediscussed.
References1. J. Maddocks, M. L. Overton, Comm. Pure and Applied Math. 48, (1995), 583–610.2. P. Freitas, M. Grinfeld, P.A. Knight, Adv. Math. Sci. Appl. 7(1), (1997), 437–448.3. A. Akay, J. Acoust. Soc. Am. 111(4), (2002), 1525–1548.4. N.M. Kinkaid, O.M. O’Reilly, P. Papadopoulos, J. Sound Vib. 267, (2003), 105–166.5. R. Krechetnikov, J. Marsden, Rev. Mod. Phys., 79(2), (2007), 519–553.6. O.N. Kirillov, Int. J. Non-Lin. Mech. 42(1), (2007), 71–87.
How to economically solve a multi-frequency problem witha nonnegative definite operator
Vladimir Druskin∗, Leonid Knizhnerman†, Mikhail Zaslavsky‡
∗‡Schlumberger-Doll Research, 1 Hampshire St., Cambridge, MA 02139, USA†Central Geophysical Expedition, Russia, 123298, Moscow, Narodnogo Opolcheniya St., house 38,
building 3∗[email protected], †[email protected], ‡[email protected]
Let us have a parametric family of equations
(A+ iωI)uω = ϕ, 0 < ωmin ≤ ω ≤ ωmax,
where A is a self-adjoint non-negative definite operator in a Hilbert space and ϕ is a nonzero vector.Given n, we wish to find basic frequencies ωj, 1 ≤ j ≤ n, such that uω can be well approximated bylinear combinations of uωj
.Some potential theory considerations invoked us to extract n/2 basic frequencies from Zolotaryov’s
approximation to ω−1/2 on a bounded positive interval and to add negative frequencies with the samemodules. The coefficients xj of linear combinations can be obtained via solution of Galerkin type
42
systems in the weighted spectral functional space:
(λ+ iω)
n∑
j=1
xj1
λ+ iωj− 1 ⊥ 1
λ+ iωk, 1 ≤ k ≤ n.
Here λ ≥ 0 represents the spectrum of A and the weight function is λ−1/2.We have experimentally found out that our choice of basic frequencies is good enough. A hypoth-
esis on the approximation error has been formulated. It is illustrated with the results of numericalexperiments.
The method of magnetic field computation in the presence of an ideal conductivemultilinked surface using the integro-differential equation of the first kind
T. V. KochubeySouthern Scientific Centre of Russian Academy of Sciences
The problem of stationary plane magnetic field computation in the presence of the ideal con-ductive surface Γ′ with the piecewise-smooth Lipschitz border has been considered. The surface isin a homogeneous medium with magnetic conductivity µ0 = const. The problem of magnetic fieldcomputation has been reduced to the following scalar boundary problem:
∆ϕ∗ = 0 outside of Γ′,
∂ϕ∗
∂n = B0n on Γ′,
∫
Γ′′
k
∂ϕ∗
∂n dΓ =∫
Γ′′
k
B0ndΓ, k = 1,N,
ϕ∗ (M) −→M→∞
0,
where B0 is an induction of non-perturbed magnetic field, ϕ∗ is a scalar potential of magnetic field
of reaction. The balance of magnetic field sources is also taken into account. Here Γ′′ =N⋃
k=0
Γ′′k
is a closure of Γ′ to closed surface Γ = Γ′ ∪ Γ′′ ∪ δΓ′′, where N is a number of internal apertures,
δΓ′′ =N⋃
k=0
δΓ′′k, δΓ
′′k is an adjacent face for Γ′ and Γ′′
k.
The ϕ∗ can be represented in the form of potential of double layer. The boundary problem hasbeen reduced in this case to the integral-differential equation of the first kind for scalar density:
Kτστ = fτ on Γ′
where
Kτστ = − 1
2π
∂
∂n
∫
Γ′
τ∂
∂n
1
rdΓ, fτ = −2B0
n.
It is possible to continue the equation on Γ if the operator which average values of functions overΓ′′
k is introduced, and following conditions are carried out:
43
fτ (M) =
fτ (M) , M ∈ Γ′,−2
mes(Γ′′
k)
∫
Γ′′
k
B0ndΓ, k = 0,N,
τ (M) =
0, M ∈ Γ′′0 ∪ δΓ′′
0 ,τ (M) , M ∈ Γ′,
ck, M ∈ Γ′′k ∪ δΓ′′
k, k = 1,N,
where ck = const and δΓ′′0 is an external border of the surface.
It was shown that the operator of the equation is linear, self-adjoint and positive in L2(Γ) thatis Hilbert space of square-integrable functions with constant value on Γ′′
k. It allowed to prove theavailability, uniqueness and stability of the solution of the equation in the operator’s energetic space[1] by applying the variational principle and Riesz’s theorem [2].
The numerical solution of the equation is carried out by constructing the minimizing Ritz’ssequence on the base of system of continuous piece-polynomial coordinate functions. This approachreduces the equation to the SLAE with the real positive-definite matrix.
The software package for the numerical realization of the developed theory has been created.The comparing of the results of the software package with the results of similar packages has shownthat the developed software package has higher accuracy and shorter time of computation.
References1. Astakhov V. I., Surface potentials and operators of potential’s theory in Dirichlet’s spaces,
Electromechanics, 2000, 2, pp. 1–18. (in Russian language)2. Mikhlin S. G., The linear equations in partial derivatives, Moscow, Higher sch., 1977. 431
p. (in Russian language)
On the connection between the regularity of matrices over idempotent semirings andthe solutions to the assignment problem
Vassili N. KolokoltsovMoscow Institute of Economics, Russia, and Department of Statistics, University of Warwick,
Coventry CV4 7AL, [email protected]
A start to (max,+)-algebra was seemingly first given in [6], where it was noted that discreteequations of dynamic programming become linear in a exotic algebra, where the real numbers areconsidered to be equipped with two distributing binary operations a⊕b = max(a, b) and a⊙b = a+b,some non-trivial results for this (max,+)-algebra were obtained and the program of its systematicstudy was put forward. As the generalized addition ⊕ is idempotent in this algebra meaning thata ⊕ a = a, this algebra became the basic example of what was later called idempotent algebra (oridempotent dioid). A couple of monographs on this algebra were published to the time when V.P.Maslov observed that this exotic linearity can be fruitfully pursued further from discrete dynamicprogramming to the differential Bellman equation of continuous time optimization theory, and theIdempotent analysis was initiated. It was baptized in [2] and its first 10 years of developmentwere summarized in monographs [3,4]. Apart from optimization theory a spectacular applicationof idempotency appeared in the theory of finite automata and formal languages. In honor of the
44
contribution to this field made by the Brazilian mathematician Imre Simon, the idempotent algebraswere started to be called tropical by some authors. The appearance and rapid development of tropicalgeometry (see e.g. reviews in [5]) gave a new twist to the development.
Let B = (bij), i, j = 1, ..., n, be a square matrix. Assignment problem consists in finding abijection F : X = 1, ..., n 7→ X maximizing the sum
∑ni=1 biF (i) among all such bijections. On the
other hand, to the matrix B there corresponds a mapping B : Rn 7→ Rn (denoted again by B withsome abuse of notation) by the formula (Bf)i = supjbij − fj. In the theory of discrete optimalcontrol and discrete event systems one often encounters the problem of solving the equation Bf = gfor a given vector g ∈ Rn. One says that the matrix B is strongly regular if there exists g such thatthe equation Bf = g has a unique solution. The long development of the theory of such equationslead eventually to the following beautiful result (obtained in [1]): B is strongly regular if an only ifthe corresponding assignment problem has a unique solution. In the aim of the present contributiona more general point of view on this topic will be presented aimed at the extension of this resultto an infinite dimensional setting, namely to the case of the state space X being a countable set.Theses results are based on the joint work of the author with M. Akian and St. Gaubert from INRIAFrance.
References1. P. Butkovic, F. Hevery, A condition for the strong regularity of matrices in the minimax
algebra, Discrete Appl. Math. 11 (1985), 209-222.2. V.N. Kolokoltsov, V.P. Maslov, Idempotent Analysis as a tool in control theory. Part
1, Funkts. Anal. i Prilosh. 23:1 (1989), 1–14. Engl. Transl. Funct. Anal. Appl.3. V.N. Kolokoltsov, V.P. Maslov, Idempotent Analysis and its Application to Optimal
Control, Moscow, Nauka, 1994 (in Russian).4. V.N. Kolokoltsov, V.P. Maslov, Idempotent Analysis and its Applications, Kluwer, 1997.5. G.L. Litvinov, V.P. Maslov (Eds.) Idempotent Mathematics and Mathematical Physics,
Contemporary Mathematics v.377, AMS, Providence, 2005.6. N.N. Vorobyev, Extremal algebra of positive matrices, Elektron. Informationsverarbeitung
und Kybernetik 3 (1967), 39-71 (in Russian).
Dessins d’enfants, generalized Chebyshev polynomials, and anti-Vandermondesystems of equations
Elena KreinesMoscow State University
This work is done under the partial financial support from the grant [email protected]
(Bicolored) dessin d’enfant is a compact connected smooth oriented surface M together with a(bipartite) graph Γ on it such that the complement M \ Γ is homeomorphic to a disjoint union ofopen discs. The theory of dessins d’enfants was initiated by A. Grothendieck in [1] and activelydeveloped thereafter. Dessins d’enfants became rather popular within the last decades; they providea possibility to describe in the easy and visually effective combinatorial language of graphs on surfacesmany difficult and deep concepts and results of Group theory, Teichmuller and moduli spaces, Maps
45
and hypermaps, Matrix models, Quantum gravity, String theory, etc. Numerous applications canbe found in [2,3] and references therein.
Bicolored dessins d’enfant are canonically related with so-called Belyi pairs, a Belyi pair is analgebraic curve X together with non-constant rational function on this curve, β : X → CP 1 withat most 3 critical values. This function is usually called the Belyi function. In the case, it isa polynomial on C, it is called a generalized Chebyshev polynomial or a Shabat polynomial . Theaforesaid correspondence is realized in the following way: if the critical values are 0, 1,∞, thenβ−1([0, 1]) is a dessin d’enfant, whose edges are β−1((0, 1)), “black” vertices are β−1(0), and“white” vertices are β−1(1). Also for a given list of valencies (degrees) of a dessin it is possibleto write down a system of algebraic equations determining the Belyi pair of this dessin. In thetalk different properties of such systems will be discussed. In particular, it appears that among thesolutions there are solutions which do not have the original list of valencies, but only a certain itssub-list. We call them parasitic solutions and say that a parasitic solution is geometric if it is aBelyi function and non-geometric otherwise. We classify all parasitic solutions. As corollaries thefollowing results can be obtained:
Theorem 1. The systems for dual dessins have “dual” parasitic solutions.
Theorem 2. Each list of valencies has at most finite number of geometric parasitic solutionsand either zero or infinitely many non-geometric parasitic solutions.
Theorem 3. Systems of equations for generalized Chebyshev polynomials have no parasiticsolutions.
In particular, it is proved that the following anti-Vandermonde system:
α1z1 + α2z2 + . . .+ αnzn = 0α1z
21 + α2z
22 + . . .+ αnz
2n = 0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
α1zn−11 + α2z
n−12 + . . .+ αnz
n−1n = 0
here αi are given positive integers and zi are unknowns,has no parasitic solutions and has exactly (n − 1)! solutions, if αi 6= αj for i 6= j.
Several other combinatorial properties of this system will be discussed.
References1. A. Grothendieck, Esquisse d’un programme, London Math. Soc. Lecture Notes Series,
Cambridge Univ. Press. 242 (1997) 5–48.2. S. K. Lando, A. K. Zvonkin, Graphs on surfaces and their applications, with an appendix
by D. Zagier, Encycl. of Math. Sciences 141, Springer, 2004.3. G. Malle, B. H. Matzat, Inverse Galois Theory, Springer-Verlag, Berlin, Heidelberg, New
York, 1999.
Preserving zeros of Jordan triple products
B. KuzmaInstitute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia.
46
A Jordan triple product, A ⋄ B := ABA is a natural product to consider on the space of Her-mitian/Symmetric/Alternate matrices, as well as on Hilbert space effects i.e., self adjoint operatorswith 0 ≤ X ≤ I.
Let F be a field and n ≥ 3. Suppose S1,S2 ⊆ Mn(F) contain all rank-one idempotents. Wewill discuss the structure of surjections φ : S1 → S2 satisfying ABA = 0 ⇐⇒ φ(A)φ(B)φ(A) = 0.Similar results are also obtained for (a) subsets of bounded operators acting on a complex or realBanach space X, (b) the space of Hermitian matrices acting on n-dimensional vectors over a skew-field D, (c) subsets of self-adjoint bounded linear operators acting on an infinite dimensional complexHilbert space. The obtained results can be applied to characterize mappings φ on matrices oroperators such that
F (ABA) = F (φ(A)φ(B)φ(A)) for all A,B,
for functions F such as the spectral norm, Schatten p-norm, numerical radius and numerical range,etc.
This is a joint work with M. Dobovisek, G. Lesnjak, C.-K. Li, and T. Petek.
Separation of variables in nonlinear Fermi equation
Yu. I. KuznetsovICM and MG SB RAS, Novosibirsk
Such characteristics of seismic activity as clustering in space, in time, in seismic energy and soon, which are observed in last decades, provide evidence of the wave nature of seismic processes.
Problem of modelling the elastic chain of rotating spherical blocks of equal size is reduced tosolving 1D nonlinear sine-Gordon equation:
∂2θ
∂ξ2− ∂2θ
∂η2= sin(θ).
Variables ξ = k0x and η = c0k0t are dimensionless coordinates, x is a coordinate in the chain, tis time, c0 and k0 are constants (in essence, characteristic velocity and wave number of the solution).
The solution has the character of a soliton. Computations based on the sine-Gordon equationhave a rather good agreement with real data corresponding to a seismic foci belt.
In this work we consider another nonlinear equation [1] with a soliton solution:
zk = (zk+1 − 2zk + zk−1)(1 + α(zk+1 − zk−1)
), (1)
z0 = zN+1 = 0, k = 1(1)N , describing N uniformly spaced equal masses fixed on a thread. Herezk = zk(t) is a deviation of k-th mass from the equilibrium state. In this system, spatial andtemporal variables can be separated. It is known [1] that the oscillatory system described by (1)has a non-linear behavior: the initial momentum energy does not dissipate in the phase space withtime, but is focused in a group of harmonics.
In this paper, by analogy to [1], we use the basis of the linear problem (1) for α = 0. However aspecial transformation is used for the coefficients of decomposition. As a result, we get an equation for
47
the Fourier coefficients which is independent from the considered harmonics. We get new conclusionsaffirming that the dynamics of discrete systems is more rich than that of continuous systems [2].
Let Z(t) = (z1(t), . . . , zN (t))T and
Z(t) =N∑
i=1
ci(t)Y(i), (2)
where Y (i) = (y(i)1 , . . . , y
(i)N )T , y
(i)0 = 0, y
(i)N+1 = 0 is a solution of the homogeneous equation
y(i)k+1 − 2y
(i)k + y
(i)k−1 = −λiy
(i)k , i = 1(1)N. (3)
Here λi is the squared frequency, Y (i) is the amplitude vector of the linear oscillator described bythe equation (1) for α = 0.
Representation (2) allows to make a transformation which leads to the following statement.
Theorem.The vector C =(c1(t), . . . , cN (t)
)Tis bound by the following relation:
C = −(I + α(BT −BH)ΛC, (4)
where I is an identity matrix, Λ = diag(λ1, . . . , λN ), λj are eigenvalues appearing in equation (3),BT is a symmetric Toeplitz matrix and BH is a persymmetric Hankel matrix:
BT =
0 t1 t2 · · · tN−2 tN−1
t1 0 t1 tN−2
t2 t1. . .
. . ....
.... . .
. . . t1 t2tN−2 t1 0 t1tN−1 tN−2 · · · t2 t1 0
,
BH =
t2 t3 · · · tN−1 tN 0t3 0 tN
. ..
tN−1... . .
. ...
tN−1 . ..
. ..
tN 0 t30 tN tN−1 · · · t3 t2
.
References1. E. Fermi, Collected papers (Note e memorie), University of Chicago Press, 1965. V. 2.2. V.K.Mezentsev, S. L. Musher, I. V. Ryzhenkova, S.K. Turitsyn, Two-dimensional
solitons in discrete systems, JETP Letters, 60 (11) (1994), 815–821.
48
On representation of the polynomials of best approximation with the weight andtheir evaluation
Lebedev V. I.Kurchatov Institute;
Institute of Numerical Mathematics of [email protected]
Let w(x), f(x) ∈ C[−1, 1] be respectively the positive weight, and the function to be approxi-mated.
We consider the classical minimax problem. To find en, the minimum of the uniform norm||(f(x) − Qn(x))w(x)||C[−1,1], among all degree n polynomials Qn(x), as well as the polynomialPn(x) where this minimum is attained. The latter polynomial is known as the best polynomialapproximation of f(x) with the weight w(x) or, briefly, PBA.
Let rn(x) = (f(x) − Pn(x))w(x), x = (x1, . . . , xn+1), xi ∈ [−1, 1], xi 6= xk, ωn+1(x, x) ==
∏n+1i=1 (x−xi), and Ln(x, x) — be the n-th degree Lagrange polynomial for f(x) with interpolation
points x. ThenPn(x) = Ln(x, y) = arg min
xmax
x∈[−1,1]|ωn+1(x, x)Ln(x, x)w(x)|.
Let x = cos θ, then the following theorem is true.Theorem. The polynomial Pn(x) is PBA if and only if there exist real E, integer m ≥ n and a
function ψ(θ) ∈ C[0, π], 0 ≥ ψ(0) > −π, 0 ≤ ψ(π) < π such that,
rn(x) = E cos((m+ 1)θ + ψ(θ)).
In this setting en = E.Zeros of rn(x) are the points
yi = cos θi, θi = ((i− 1/2)π − ψ(θi))/(m+ 1), i = 1,m+ 1.
The representation of the error of approximation in trigonometric form in terms of the phasefunction ψ(θ) is the basis of the iterative method.
The interpolation quadrature formula for the singular integrals with Hilbert kernel is used forthe corrections of the phase function. This formula together with the formula for the zeros of theerror determines the algorithm of collective dynamics of zeros and e-points when the weight functionw(x) is disturbed.
The iterative methods for evaluation of the PBA parameters based on the methods of inverseanalysis, perturbation theory and asymptotic formulae are formulated. Numerical experiments showthe high efficiency of the proposed approach.
References1. Lebedev V. I., Proc. Lobachevskii Math. Inst., v. 20, Kazan’, 2003, 37–86.2. Lebedev V. I., Sb. Math. v. 195, 10, 2004.
49
Fast numerical solution methods for two-dimensional Fredholm integral equations ofthe second kind
Fu-Rong Lin∗, Wen-Jing Xie†, Feng Liang‡
∗†‡Department of Mathematics, Shantou University, Shantou Guangdong 515063, China∗[email protected]
In this paper we consider numerical solution methods for two dimensional Fredholm integralequation of the second kind
f(x, y)−∫ β
α
∫ β
αa(x, y, u, v) f(u, v) dudv = g(x, y), (x, y) ∈ [α, β]× [α, β],
where a(x, y, u, v) is smooth and g(x, y) is in L2[α, β]2. We discuss polynomial interpolation meth-ods for four-variable functions and then use the interpolating polynomial to approximate the kernelfunction a(x, y, u, v). Based on the approximation we deduce fast matrix-vector multiplication algo-rithms and efficient preconditioners for the above two dimensional integral equations. The residualcorrection scheme is used to solve the discretization linear system. Numerical results are given toillustrate the efficiency of our algorithms.
Dequantization of linear operators
G. L. Litvinov∗, G. B. ShpizIndependent University of Moscow
For functions defined on Cn (or Rn or Rn+) the well-known Maslov dequantization generates the
following dequantization transform:
f 7→ f(x) = limh→0
h log(| f(exp(x/h) |) = lims→+∞
(1/s) · log(| f(exp(sx) |),
where x ∈ Rn, and h, s = 1/h are real parameters. If f is a polynomial, then the subdifferentialof f at the origin coincides with the Newton polytope of f ; it is possible to generalize this resultto a wide class of functions and convex sets (see, e.g., [1] for details). The Maslov dequantizationgenerates a passage from the traditional mathematics to the so-called tropical mathematics.
Our aim is to apply the dequantization transform to matrix elements of operator semigroupsgenerated by linear operators.
Suppose that S is a semigroup and s 7→ πs is a linear representation of S in a complete (orquasicomplete) barreled locally convex space (by continuous operators). Denote by V ′ the dual spaceto V and by 〈v′, v〉 the value of a functional v ∈ V ′ on an element v ∈ V . If s 7→ πv′,v(s) = 〈v′, πsv〉is a matrix element of π, then its dequantization πv′,v is defined by the formula:
πv′,v = lims→∞(1/s) · log(| 〈v′, πsv〉 |).
We discuss the cases S = R+ or S = Z+. If v and v′ are fixed, then πv′,v ∈ S ∪ ∞.
50
Proposition 1. Let A be a linear operator in V , πs = exp(sA), and dimV <∞. Then the set ofall dequantizations πv′,v coincides with the set of real parts of all eigenvalues of A.
There are generalizations of this result for the case dimV =∞.
Suppose that for every v′ ∈ V ′ there exists a number r > 0 such that the set r−s · (| 〈v′, πsv〉 |), s ∈ S is bounded for every v ∈ V and the number r does not depend on v′ ∈ V ′. Then therepresentation π is called exponential. Note, that if V is a Banach space and π is weakly continuous,then π is exponential. In the general case the spectral radius ρπ of π is defined by the formula:
ρπ = infr | r−sπsv → 0 weakly for every v ∈ V as s→ +∞
Proposition 2. If A is a bounded linear operator in a Banach space V , S = Z+, π = As, thenρπ = ρ(A) = lims→∞ ‖As‖1/s, i.e. ρπ is the traditional spectral radius of A.
Theorem 1. If π is exponential, then
log ρπ = supπv′,v | v′ ∈ V ′, v ∈ V .
Theorem 2. Suppose that A is a compact operator and πs = As, where s ∈ S = Z+. Then the setπv′,v of all dequantizations of π coincides with the set of all numbers of the form log(| λ |), whereλ runs the spectrum of A.
The work is supported by the RFBR-CNRS grant 05-01-02807.
References1. G. L. Litvinov, Maslov dequantization, idempotent and tropical mathematics: A brief intro-
duction. J. Math. Sci., 2007, vol. 140, 3, pp. 426–444.
Linear methods and error autocorrection in rational approximation
G. L. LitvinovIndependent University of Moscow
Let ϕ0, ϕ1, . . . , ϕn and ψ0, ψ1, . . . , ψm be two collections consisting of linearly independentfunctions of an argument x belonging to some (possibly multidimensional) set X. Consider theproblem of constructing an approximant of the form
R(x) =a0ϕ0 + a1ϕ1 + · · ·+ anϕn
b0ψ0 + b1ψ1 + · · ·+ bmψm=P (x)
Q(x)(1)
to a given function f(x) defined on X. Let an “abstract” construction method for the approximant ofthe form (1) be linear in the sense that the coefficients of (1) can be determined from a homogeneoussystem of linear algebraic equations
Hy = 0, (2)
where H is a matrix of dimension (m+ n+ 2)× (m+ n+ 1) and y = a0, a1, . . . , an, b0, b1, . . . , bm.Of course, an additional condition (e.g., of the type b0 = 1, or bm = 1, or a0 = 1) can be added.
51
Pade approximations, linear Pade–Chebyshev approximations etc. (and even best approximations)could be treated by this way (see, e.g., [1]). The system of equations can be very ill-conditioned andit can lead to very significant errors in the approximant coefficients. However, these errors do notaffect the accuracy of the approximant (the error autocorrection effect, see, e.g., [1]).
Let the coefficients ai, bj give an exact or an approximate solution of (2) and let ai, bj give
another approximate solution obtained in the same way. Set ∆ai = ai − ai, ∆bj = bj − bj;these errors arise due to perturbations of the approximated function f(x) or due to calculationerrors. Set ∆P (x) =
∑ni=0 ∆aiϕi, ∆Q(x) =
∑mj=0 ∆bjψj . It is easy to see that the vector
∆y = ∆a0,∆a1, . . . ,∆an, ∆b0, . . . ,∆bm is an approximate solution of (2). So the function ∆P∆Q
can be treated as an approximant to the same function f(x).The following uncertainty relation is valid:
∆(P
Q) = δQ(
∆P
∆Q− P
Q) ≈ δQ · (∆P
∆Q− f) ∼ ε,
where δQ = ∆QQ+∆Q is the relative error of Q, the difference ∆P
∆Q − f is the absolute error of the
approximant ∆P∆Q to the function f , and ε is the absolute “theoretical” error of our method; the
argument x can be treated as fixed.
The error autocorrection effect means that in a calculation all the intermediate calculation errorscompensate each other, so the final result is much more accurate than the intermediate results. Inthis case standard interval estimates (e.g., in the framework of Yu. Matijasevich’s “a posterioriinterval analysis”) are not realistic, they are too pessimistic. The error autocorrection effect appearsin some popular numerical methods, e.g., in the least squares method.
References1. G. L. Litvinov, Error autocorrection in rational approximation and interval estimates. Cen-
tral European Journal of Mathematics, vol. 1,1, 2003, p. 36–60. See also arXiv:math.NA/0207183(http://arXiv.org).
Overcoming principal uncertainties in calibration of matrix population models
Dmitrii O. LogofetInstitute of atmospheric physics RAS,
Moscow State University, Mechanics and Mathematics [email protected]
Matrix models of stage-structured population dynamics take on a general form of differenceequations,
x(t+ 1) = Ax(t),
where x(t) denotes the n-vector of population structure, i.e., the vector of abundances of n stage-specific groups of individuals within a population, and the projection matrix A consists of stage-specific vital rates, the matrix pattern reflecting a variety of life cycles for individuals of a speciesunder study. When the projection matrix of a model for perennial plants, like, e.g., raspberry Rubusidaeus colonizing forest clear-cut areas, is to be calibrated on data, the very sense of the vital rates
52
suggests monitoring of marked individual plants during successive years, thereafter calculating therates directly from the data. This can be effectively done for stage-specific survival and ontogenetictransition rates. But the rates of reproduction bear principal uncertainty as far as the parent plantscan hardly be determined for the (not yet marked!) recruitment.
To overcome this insurmountable uncertainty in data and to conclude the model calibration, weapply a general principle that suggests substitution an adaptation conjecture (“empirical general-ization”) for the lack of data/information. This principle of adaptive optimality implies that a setof the stage-specific reproduction rates can be found such that maximizes the overall populationgrowth rate (the dominant eigenvalue of matrix A) under certain constraints on the rates whichensue from expert knowledge of population biology of the species in hand and from the data. Theoriginal constrained optimization problem is of a nonlinear kind, with another principal uncertaintyin the local-vs.-global character of a solution to be found by a numerical routine. To overcome thiskind of uncertainty, we substitute maximization of the potential growth indicator function for theoriginal problem. The latter reduces to a linear programming problem, whose global solution canalways be found.
Calculation of the Schein rank for Boolean matrices
E. E.MarenichMurmansk State Pedagogic University
Let P = 0, 1 be a two-element Boolean lattice, Pm×n a set of all m× n matrices over P , andA ∈ Pm×n.
According to K.X. Kim [1], the definition of the Schein rank over a Boolean lattice P = 0, 1was given by Schein. For definition of the Schein rank over semirings we refer to K.X. Kim andF.W. Roush [2]. Calculation methods for the Schein rank over chains was considered in [3], [4] byDi Nola and S.Sesa.
Let Γ = Γ(V1 ∪ V2, E) be a bipartite graph, and let U be a finite set.A function f : V1 ∪ V2 → 2U is called an U -subsets coding function for Γ, or more simply, coding
function, if for any vertices v1 ∈ V1, v2 ∈ V2 the conditions v1, v2 ∈ E and f(v1) ∩ f(v2) 6= ∅ areequivalent. We call f(v) the code of v ∈ V . Note that there exist coding functions for any bipartitegraph Γ.
Denote by S the family of all finite U such that there exist U -subsets coding functions for Γ.The intersection number of the bipartite graph Γ is
nint∅(Γ) = minU∈S|U |.
A clique in a bipartite graph Γ is any maximal complete bipartite subgraph containing at leastone edge.
Let Γ and Γ1, . . . ,Γk be a bipartite graph and a family of its subgraphs. We say that subgraphsΓ1, . . . ,Γk cover Γ if each vertex and each edge of Γ are contained in at least one subgraph.
In the following theorem we show that intersection number calculations for any bipartite graphΓ can be reduced to counting the number of cliques covering the graph.
Theorem 1. Let Γ = Γ(V1 ∪ V2, E) be a bipartite graph without isolated vertices. Then theintersection number of Γ is equal to the least number of cliques covering Γ.
53
Let Γ′ be obtained from Γ by deleting all isolated vertices. Then nint∅(Γ) = nint∅(Γ′).
For any matrix A there exists an associated bipartite graph Γ(A) = Γ(V1 ∪ V2, E) such thatV2 = 1′, 2′, . . . , n′, i, j′ ∈ E iff aij = 1.
The following theorem reduces the Schein rank problem for any matrix A to the intersectionnumber calculation for the associated bipartite graph Γ(A).
Theorem 2. Let P = 0, 1 and A ∈ Pm×n be a two-element Boolean lattice and a matrix.Then the Schein rank of A is equal to the intersection number of the associated bipartite graphΓ(A).
Corollary 1. Let P = 0, 1 and A ∈ Pm×n be a two-element Boolean lattice and a matrix.Then the Schein rank of A is equal to the least number of cliques covering Γ′, where Γ′ is obtainedfrom the associated bipartite graph Γ(A) by deleting all isolated vertices.
From Corollary 1 it obviously follows that the Schein rank of the identity matrix E = En×n ∈Pn×n equals n. The following corollary gives the criterion for A ∈ Pm×n to be a CR-matrix.
Corollary 2. Let P = 0, 1 be a two-element Boolean lattice, A ∈ Pm×n a nonzero matrix,and Γ(A) the associated bipartite graph for A. The following conditions are equivalent:i) The Schein rank of A is equal to 1.ii) Γ′ is a complete bipartite graph, where Γ′ is obtained from Γ(A) by deleting all isolated vertices.
Theorem 2 reduces the Schein rank calculation for any matrix A to the calculation of the leastnumber of cliques covering the graph, i.e., to the problem of minimum covering. One more applicationof Theorem 1 follows.
Theorem 3. Let P = 0, 1 and E = En×n ∈ Pn×n be a two-element Boolean lattice and theidentity matrix. Let E denote the complement of E. Then the Schein rank of E is equal to theminimal k such that
n ≤(k[k/2]
).
This research is conducted in accordance with the Thematic plan of Russian Federal EducationalAgency, theme 1.03.07.
References1. Kim K.X., Boolean matrix theory and applications, Marcel Dekker, New York, 1982.2.Kim K.X., Roush F.W., Generalized fuzzy matrices, Fuzzy Sets Systems 4 (1980), 293–315.3.Di Nola, Sessa S., On the Schein rank of matrices over linear lattice, Linear Algebra Appl.
118(1989), 155–158.4.Di Nola, Sessa S., Determining the Schein rank of matrices over linear lattice and finite
relational equations, The Journal of Fuzzy Mathematics, vol. 1, No. 1, 1993, 33–38.
Lattices of matrix rows and matrix columns
V. E. MarenichMurmansk State Pedagogic University
The following notations will be used throughout.Denote by (P, ∧, ∨,≤) a lattice. Let Pm×n be a set of all m×n matrices over P and A ∈ Pm×n.
54
If for given elements a, b ∈ P the greatest solution of the inequality a ∧ x ≤ b exists, then it isdenoted by
⟨ba
⟩P
=⟨ba
⟩, and is called the relative pseudocomplement of a in b. If
⟨ba
⟩exists for all
a, b ∈ P then (P, ∧, ∨,≤) is called a Brouwerian lattice.The partially ordered set (Pm×1,≤) is a lattice with meet and join operations denoted by ∧ and
∨. Any partially ordered set is called, more simply, a poset.Let A = ||aij || ∈ Pm×n. We call A(i) = (ai1, . . . , ain) the i-th row vector andA(j) = (a1j , . . . , amj)
t
the j-th column vector. The linear span of row vectors is called a row subspace for A and is denotedby Row(A). The linear span of column vectors is called a column subspace for A and is denotedby Column(A). Define posets: (Column(A),≤) with respect to the partial order ≤ induced bythe lattice (Pm×1,≤) and (Row(A),≤) with respect to the partial order ≤ induced by the lattice(P 1×n,≤). The poset (Column(A),≤) is an upper semilattice, in which join ∨ coincides with thejoin operation ∨ in the lattice (Pm×1,≤),
u∨v = u+ v = u ∨ v, u, v ∈ Column(A).Suppose (P, ∧, ∨,≤) is a complete distributive lattice. If for any subset S, S ⊆ Column(A),
there exists the least upper bound ∨S ∈ Column(A), then the poset (Column(A),≤) is a completelattice, in which join ∨ coincides with the join operation ∨ in the lattice (Pm×1,≤). If (P, ∧, ∨,≤)is a finite distributive lattice, then (Column(A),≤) is a lattice.
In some cases we can express the meet operation ∧ in the lattice (Column(A),≤) by the followingformulas.
Theorem 1. Let (P, ∧, ∨,≤) be a lattice, A ∈ Pm×n.i) If (P, ∧, ∨,≤) is a complete Brouwerian lattice, then (Column(A),≤) is a complete lattice withmeet and join operations, ∧ and ∨, where
∧S = A⟨∧SA
⟩, S ⊆ Column(A).
ii) If (P, ∧, ∨,≤) is a Brouwerian lattice, then the poset (Column(A),≤) is a lattice with meet andjoin operations, ∧ and ∨, where
u∧v = A⟨u∧v
A
⟩= A
(⟨uA
⟩∧
⟨vA
⟩), u, v ∈ Column(A).
iii) If (P, ∧, ∨,≤) is a Boolean lattice, then the poset (Column(A),≤) is a lattice with meet andjoin operations, ∧ and ∨, where
u∧v = A · tA · (u+ v), u, v ∈ Column(A).
Sometimes we can express the meet operation ∧ more simply.Theorem 2. Let (P, ∧, ∨,≤) be a Brouwerian lattice and matrix A ∈ Pn×n an idempotent.
Theni) the meet operation ∧ in the lattice (Column(A),≤) is given by
u∧v = A (u ∧ v), u, v ∈ Column(A);ii) (Column(A),≤) is a distributive lattice.
Note that a similar theorem holds for the lattice (Row(A),≤).A matrix A is called a regular matrix if there exists B ∈ Pn×n such that ABA = A. According [1],
A is a regular matrix whenever there exists an idempotent C such that Column(A) = Column(C).For the following result we refer to K.A. Zaretski, [2].Theorem (Zaretski). Let P = 0, 1 and A ∈ Pn×n be a two-element Boolean lattice and a
matrix. Then (Column(A),≤) is a distributive lattice whenever A is a regular matrix.
Zaretski’s theorem can be extended to Brouwerian lattices.Theorem 3. Let (P, ∧, ∨,≤) be a Brouwerian lattice, A ∈ Pn×n a regular matrix, and C ∈
Pn×n an idempotent such that Column(A) = Column(C). Theni) the meet operation ∧ in the lattice (Column(A),≤) is given by
u∧v = C (u ∧ v), u, v ∈ Column(A),ii) (Column(A),≤) is a distributive lattice.
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For semigroups of binary relations, the statement ii) of Theorem 3 was proved by K.A. Zaretski,see [2]. The similar result for fuzzy lattices was proved by K.X. Kim and F.W. Roush, see [3].
This research is conducted in accordance with the Thematic plan of Russian Federal EducationalAgency, theme 1.03.07.
References1. K. H. Kim, Boolean matrix theory and applications, Marcel Dekker, New York, 1982.2. K. A. Zaretski, Regular elements in the semigroup of binary relations, Uspekhi mat. nauk,
17(3): 105–108, 1962.3. K. X. Kim, F. W. Roush, Generalized fuzzy matrices, Fuzzy Sets Systems 4 (1980), 293–315.
Matrix algebras and their length
O. V. MarkovaMoscow State University, Mathematics and Mechanics Dept.
Let F be a field and let A be a finite-dimensional F-algebra. A number l(S) is called a length ofa finite generating set S of this algebra provided it equals the smallest number k, such that wordsof the length not greater than k generate A as a vector space. The length of the algebra A, denotedby l(A), is the maximum of lengths of all its generating sets.
The problem of evaluating the length of the full matrix algebra in terms of its order was posedby A. Paz in [3] and has not been solved yet. Some known upper bounds (see [3,2]) are not linear.
Using our results [1] on the length of upper-triangular matrix algebra and on the upper andlower bounds for the lengths of direct sums of algebras, we have evaluated lengths of certain upper-triangular matrix subalgebras.
The linear upper bound for the length of commutative subalgebras of Mn(C) was obtained in[3]. We have generalized this result on commutative matrix subalgebras over an arbitrary field. Alsowe have obtained the following characterization of commutative matrix subalgebras of the maximallength:
Theorem 1. Let F be an algebraically closed field. Commutative subalgebra R in Mn(F) is of themaximal length if and only if it is generated by a non-derogatory matrix.
A positive answer has been obtained to the question whether the length of an algebra can besmaller than the length of its subalgebra. That is, for any natural number k and for an arbitraryfield F there exist such F-algebra A in upper triangular matrix algebra and its subalgebra A′, thatl(A′)− l(A) = k.
I would also like to thank my supervisor Dr. A. E. Guterman for the statement of the problemand for useful discussions.
References1. O. V. Markova, On the length of upper-triangular matrix algebra. Uspekhi Matem.
Nauk, 60(2005), no. 5, 177–178. [in Russian]; English translation: Russian Mathematical Surveys,60(2005), no. 5, 984–985.
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2. C. J. Pappacena, An Upper Bound for the Length of a Finite-Dimensional Algebra, Journalof Algebra, 197(1997), 535–545.
3. A. Paz, An Application of the Cayley–Hamilton Theorem to Matrix Polynomials in SeveralVariables, Linear and Multilinear Algebra, 15(1984), 161–170.
A fast algorithm for computing the nullspace of some structured matrices
Nicola Mastronardi∗, Marc Van Barel†, Raf Vandebril‡∗Istituto per le Applicazioni del Calcolo “M. Picone” Bari, Italy
†‡Katholieke Universiteit Leuven, Department of Computer Science, Belgium∗[email protected], †[email protected],
The problem of computing the nullspace of (block) Toeplitz/Hankel matrices arises in differentapplications, such as system theory, linear prediction and parameter estimation. Many algorithmshave already been available in the literature to compute the nullspace of the latter structured matri-ces. In this talk an algorithm based on the generalized Schur algorithm will be presented. Similaritiesand differences with the existing algorithms will be also emphasized.
Algebraic Multigrid Methods and Block Preconditioning for MixedElliptic Hyperbolic Linear Systems,
Applications to Stratigraphic and Reservoir Simulations
Roland Masson∗, Y. Achdou†, P. Bonneau‡, P. Quandalle§∗‡§Institut Francais du Petrole,
†University Paris VII∗[email protected]
In multiphase flow in porous media, and stratigraphic modeling, the linear systems obtained afterfinite volume discretization in space, fully implicit time integration and Newton type linearizationcouple basically an elliptic/parabolic variable (say the pressure) with hyperbolic variables (say thecompositions or saturations). For such systems, unknown based preconditioners [1,2,3] allow toadapt efficiently the preconditioner to each type of variable using algebraic multigrid [4] for thepressure equations and e.g. incomplete factorizations for the remaining variables. The couplingbetween the variables is obtained by multiplicative combination of both the multigrid and incompletefactorization preconditioners.
In this talk, such methods will be detailed and results will be presented on reservoir simulation,basin simulation, and stratigraphic modeling examples.
In reservoir simulation, implicit well equations and high compressibility lead to strongly nondiagonal dominant additional equations for the pressure block which usually have a bad effect on
57
the performance of algebraic multigrid methods. We will discuss several solutions to recover theefficiency of the combinative method in such cases.
References1. S. Lacroix, Y.V. Vassilevski, and M.F. Wheeler, Decoupling preconditioners in the Im-
plicit Parallel Accurate Reservoir Simulator (IPARS), Numerical Linear Algebra with Applications8 (2001) pp. 537–549.
2. R. Scheichl, R. Masson, J. Wendebourg, Decoupling and block preconditioning forsedimentary basin simulations, Computational Geosciences 7, pp. 295–318, 2003.
3. Y. Achdou, P. Bonneau, R. Masson, P. Quandalle, lgebraic Multigrid Based Precon-ditioners for Oil Reservoir Simulation, ECMOR X Proceedings, Amsterdam, 2006.
4. J.W. Ruge and K. Stuben, Algebraic Multigrid (AMG), In Multigrid Methods (S.F. Mc-Cormick, ed.), Frontiers in Applied Mathematics, vol. 5, SIAM, Philadelphia, 1986.
On the sum of elements of a nonnegative two-by-two matrix
Jorma K. MerikoskiDepartment of Mathematics, Statistics and Philosophy, FI-33014 University of Tampere, Finland
Let A be a nonnegative two-by-two matrix with row sums r1 and r2, and column sums c1 andc2. Let m be a nonnegative integer, and let suB denote the sum of elements of a matrix B. Drury,Merikoski, Laakso and Tossavainen [Electron. J. Linear Algebra 10 (2003), 280–290] proved that ifm is even, then suAm ≤ (r1c1)
m2 +(r2c2)
m2 . The proof is surprisingly hard. Some further questions,
motivated by this inequality, are discussed.
Application of the multigrid method to solving diffusion-type equations
M. E. Ladonkina, O. Yu. Milyukova, and V. F. TishkinInstitute for Mathematical Modeling, Russian Academy of Sciences Miusskaya sq. 4a, Moscow,
125047 [email protected]
Numerical modeling of many problems of mathematical physics, such as gas dynamic or hydrody-namic problems, is impossible without taking into account diffusion processes. Describing diffusionprocesses involves solving parabolic equations. The application of explicit schemes to approximateparabolic-type equations imposes strong constraints on the time step. The application of implicitschemes removes these constraints, but it requires solving large systems of linear algebraic equations,which renders using such schemes unexpedient.
We suggest a difference algorithm for solving the initial-boundary value problem for the heatequation
cv∂T
∂t= div k gradT (1)
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on the basis of the multigrid method [1]. In (1), T is the temperature, cv is the coefficient of heatcapacity, and k is the heat conduction.
The value of the grid function at the (n+ 1)th time layer is determined as follows [2]. First, oneor several simple smoothing iterations for the equations representing the purely implicit scheme ona fine grid with step h
Ahuh = fh. (2)
are performed. The discrepancy thus obtained is projected onto a coarse grid with step H = 2h, onwhich the equation for corrections
AH∆H = RH (3)
is solved; here, RH is the discrepancy on the coarse grid. The correction ∆H is then interpolatedonto the fine grid and used to evaluate the grid function at the (n+ 1)th time layer by the formulaun+1 = us
h − δh, where δh is the correction on the fine grid and us is the solution obtained at thesmoothing stage. Thus, only one iteration of the two-grid cycle is performed.
Since solving equation (3) on the coarse grid requires much fewer operations than solving equation(2) and, moreover, only one or several smoothing iterations are performed, the suggested methodfor solving the multidimensional heat equation requires less computational time than the implicitscheme on the fine grid (that is, solving equation (2)).
The stability and accuracy of the suggested algorithm were theoretically studied for model prob-lems by using Fourier analysis. For a one-dimensional model problem, the absolute stability of themethod with respect to the initial data and the right-hand side was proved [2]. For a two-dimensionalmodel problem, it was shown that if τ = hβ and 0 < β < 2, then ‖ uh−un+1 ‖L2= O((h2)h2β(1−3.5δ)),where τ is the time step and 0 < δ < 1/3.5 is an arbitrary given real number. Thus, the error‖ uh−un+1 ‖L2 has higher order than ‖ uh−uT ‖L2 , where uT is the projection of the exact solutionto equation (1) onto the fine grid, and the suggested algorithm makes it possible to obtain a verygood approximation to the exact solution by using the implicit scheme on the fine grid.
The theoretical results were confirmed by computations of model problems, including two-dimen-sional ones. Computations of model problems with discontinuous heat conduction also demonstratedthe stability and good accuracy of the method.
References1. Fedorenko R. P., A relaxation method for solving difference elliptic equations, Zh. Vychisl.
Mat. Mat. Fiz. 1961. Vol.1. 5. P.922–927.2. Ladonkina M. E., Milyukova O. Yu., Tishkin V. F., A numerical algorithm for diffusion-
type equations based on the multigrid methods, Mat. Model. 2007. Vol.19. 4. P.71–89.
New Smoothers in Multigrid Methodsfor Strongly Nonsymmetric Linear Systems
Evgenia M. Andreeva, Lev A. Krukier, Galina V. MuratovaSouth Federal UniversityRostov on Don, Russia
This research is supported by the Russian Foundation for Basic Research (RFBR) under theGrants No. 06-01-39002, 06-01-00038.
59
Multi-grid methods (MGMs) are fast iterative solvers based on the multilevel or the multi-scaleapproach. A typical application of the multigrid method is to numerical solution of partial differentialequations in two or higher dimension. Alternatively, the MGM can be applied in combination withany of the common discretization techniques; in such case, it is among the fastest solution techniquesknown today.
Multigrid method does not mean only a fixed multigrid algorithm applying to all boundaryvalue problems in the general setting, and it may usually refer to an algorithmic framework of themultigrid techniques. The efficiency of the MGM heavily depends on the adjustment of the involvedcomponents, e.g., the basic linear solver or the smoothing procedure.
We consider new smoothers resulted from a special class of triangular skew-symmetric splitting it-eration methods for the multigrid methods used to solve the systems of linear equations with stronglynonsymmetric coefficient matrices, which may be produced by the central-difference approximationof a convection-diffusion equation with dominant convection.
In specific, we consider the system of linear equations
Ay = f, (1)
where A is a large sparse nonsymmetric matrix. Everywhere we assume that the matrix A is positivereal, i.e., its symmetric part A0 = 1
2 (A+AT ) is positive definite. Here, we denote by A = A0 +A1,with A0 = 1
2(A+AT ) = AT0 > 0 and A1 = 1
2 (A−AT ) = −AT1 . Note that A1 is the skew-symmetric
part of the matrix A. Besides, we assume that the matrix A is strongly nonsymmetric, i.e., it holdsthat ‖A0‖⋆ ≪ ‖A1‖⋆ in the sense of certain matrix norm.
We solve the linear system (1) by the multigrid methods with the smoothers resulted from thetriangular skew-symmetric splitting iteration methods TKM1 and TKM2 described as follows:
Byn+1 − yn
τ+Ayn = f, (2)
TKM1: B = αE + 2Kℓ or B = αE + 2Ku, (3)
TKM2: B = αiE + 2Kℓ or B = αiE + 2Ku, (4)
where τ is an iterative parameter, and Kℓ and Ku are, respectively, the strictly lower- and upper-triangular parts of the skew-symmetric matrix A1.
We remark that TKM1 is the triangular skew-symmetric splitting iteration method withoutparameter, and TKM2 is a suitable modification about it. These two splitting iteration methods areapplicable to linear systems whose coefficient matrices may be not diagonal dominant.
We suggest the following formulas for choosing the parameters involved in both TKM1 andTKM2:
α = ‖M‖ , αi =∑n
j=0 |mij |, i = 0, n,
where M = (mij) := A0 +Ku −Kℓ is the symmetric matrix.The convergence of the multigrid methods with the TKM1 and the TKM2 smoothers are proved
[2], and some theoretical and numerical results about the convergence of the MGM with differentsmoothers are also presented.
Acknowledgments: The authors are very much indebted to Prof. Zhong-Zhi Bai for his usefulcomments and suggestions, in particular, for his kind invitation of visiting State Key Laboratoryof Scientific/Engineering Computing, Academy of Mathematics and Systems Science, of ChineseAcademy of Sciences, P.R. China, during August-September in 2007, under the support of China-Russia (NSFC-RFBR) International Cooperative Research Project.
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References1. L. A. Krukier, L. G. Chikina, T. V. Belokon, Triangular skew-symmetric iterative
solvers for strongly nonsymmetric positive real linear system of equations, Appl. Numer. Math.,41(2002), 89–105.
2. G. V. Muratova, E. M. Andreeva, Multigrid solver with different smoothers for convection-diffusion problem, in Proceedings of International Summer School “Iterative Methods and MatrixComputations”, RSU publisher, Rostov on Don, (2002), pages 437–443.
Stabilization of finite-difference scheme for Stokes problem
M. Olshanskii, E. MuravlevaDept. Mechanics and Mathematics, Moscow State University,
Leninskije gory 1, Moscow 119899, [email protected]
We present a new stabilized method for the Stokes problem. Staggered grids (components of thevelocity field are computed in grid’s crosspoints and pressure — in the cells’ centers) are considered.We deal with second order discrete gradient operator which leads to high precision results. Ithas bidimensional kernel in 2D case and mesh-dependent kernel in 3D case (number of spuriouspressure modes linearly depends on the quantity of discretization points). Due this fact LBB stabilitycondition violates. Our method is independent of the space dimension and does not require choosingany mesh-dependent parameters. It always leads to symmetric linear systems. The new methodis unconditionally stable and has simple and straightforward implementations. The stabilizationstrategy relies on projection operator whose action can be evaluated locally at the mesh level usingonly standart nodal data structures. As a result, an existing code can be easily modified to handlethe stabilization procedures. So, we solve the system
(A BBT C
)(up
)=
(fg
)
where C is a sparse matrix composed of number of blocks, and each block is associated with anynode. Numerical experiments were held not only for traditional viscous fluid, but also for morecomplicated non-newtonian models (Bingham fluids).
On a class of Singular Structured Matrices with Integer Spectra
Tatjana Nahtman∗, Dietrich von Rosen†
∗Institute of Mathematical Statistics, University of Tartu, Estonia†Department of Biometry and Egineering, Swedish University of Agricultural Sciences
The objective of this paper is to consider a class of structured singular nonsymmetric matriceswith integer spectrum which has applications in statistical sampling theory. The spectrum of the
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matrices, and right and left eigenvectors are presented. The matrices of eigenvectors are connected toVandermonde matrices. We shall also discuss several interesting properties of the class of matrices,for example, several triangular factorizations.
Operator equations for eddy currents on singular carriers
J. A. NaumenkoSouthern Scientific Centre of Russian Academy of Science, Rostov-on-Don, Russia
There is a number of various practical problems that needs to compute numerically magneticfields in the medium with the multiconnected crack.
Let’s consider the space L2(S) that consists of two-component complex square-integrable onS vector functions. We suppose that the multiconnected surface S and its boundary satisfy the
Lipschitz’s conditions. The space L2(S) can be decomposed to the sum: L2(S) = L(C)2 (S) ⊕
L(Γ)2 (S)⊕L(Π)
2 (S). Here L(Π)2 (S) consists of potential fields generalized by Weyl [1], L
(C)2 (S) consists
of generalized solenoidal fields and L(Γ)2 (S) consists of generalized harmonic fields. Let’s designate
L = L(C)2 (S)⊕L(Γ)
2 (S). We use the orthoprojector P = PLPS below where PS vanishes normal to Sfield component and PL is orthoprojector L2(S)→ L.
We consider the following operator equations on S:
Kσ = f1 (1)
δ = λKδ + f2. (2)
δ = λ∂
∂tKδ + f3(t). (3)
Here K = PΓ, Γξ = 14π
∫∫S
ξrNM
dSN , λ is some imaginary parameter, f1, f2, f3(t) ∈ L.
We proof the existence, uniquiness and numerical stability of the equation (1) solution in the
energy space LK of operator K in the most useful case f1 ∈ B1/22 (S) [2,3]. Here B
1/22 (S) is the Besov
space [4]. The existence, uniquiness and numerical stability of the equation (2) solution in L areproofed by the theory of Fredholm. f2 ∈ L is enough. The similar properties of the equation (3)solution is proofed by the spectral method.
The operator K is linear, sel-ajoint and positive. This propeties open broad possibilities foreffective numerical solving of equations (1)–(3).
The software package was built on the basis of the described theory. Some practical problems ofelectrodynamics were computed by it.
References1. Weyl H., The method of orthogonal projection in potential theory, Duke Math. J. 1940. V. 7.
P. 411–444.2. Naumenko J. A., Astakhov V. I., Mathematical modeling of magnetic field in presence of
ideal conducting surface with boundary, Electromechanics, 2003. 5. P. 11–16.
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3. Naumenko J. A., Local attribute of correctness and numerical computing of integral equationof first kind for density of eddy currents, The bulletin of the Southern Science Centre, 2005. 4. P.3–8.
4. Besov O.V., Il’in V.P., Nikolsky S. M., Integral representations of functions and embed-ding theorems, Moscow, Nauka, 1996. 480 p.
Augmented Lagrangian approach for saddle-point problems with indefinite(1, 1)-block
Maxim A. OlshanskiiLomonosov Moscow State [email protected]
We discuss block preconditioners for a system of linear algebraic equations of saddle-point typehaving indefinite and not necessarily symmetric (1, 1) block. The approach is based on consideringan augmented system and a special multilevel method as an inner solver for a preconditioned Krylovsubspace method. To explain convergence properties of the method we first prove bounds on theeigenvalues of the augmented (1, 1) block. Next we consider the application of the method to solve aproblem arising in the linear instability analysis for the Navier-Stokes equations. Several new resultsare proved for the Schur complement of the linearized Navier-Stokes equations. This provides furtherinsight into the convergence properties of the approach.
Quasiseparable matrices/polynomials and Lipschitz stability of Jordan bases
Vadim OlshevskyUniversity of Connecticut, [email protected]
This talk consists of two parts that are not related to each other.
(i) In the first part several recent algorithms that exploit properties of matrices having specialquasiseparable structure will be described. In particular, several generalizations of the Bjorck-Pereyra and Traub algorithms for quasiseparable-Vandermonde matrices will be presented.An interpretation via signal flow graphs will be described to fully characterize the classesof quasiseparable, semiseparable and well-free matrices. Joint work with Tom Bella, YuliEidelman, Israel Gohberg and Eugene Tyrtyshnikov.
(ii) In the second part we study first partial stability of the Jordan chains matrices under generalperturbations. Under certain conditions, Lipschitz-type bounds are obtained. Secondly, westudy Jordan-structure-preserving perturbations of matrices selfadjoint in the indefinite innerproduct. The main result is Lipschitz stability, under small perturbations, of canonical Jordanbases (i.e., eigenvectors and generalized eigenvectors enjoying a certain flipped orthonormality
63
relation) of matrices selfadjoint in the indefinite inner product. The proof relies upon theanalysis of small perturbations of invariant subspaces, where the size of a perturbation of aninvariant subspace is measured using the concepts of a gap and of a semigap. Joint work withTom Bella and Upendra Prasad.
A new class of low-tensor rank matrices closed under inversion
Ivan Oseledets∗†, Eugene Tyrtyshnikov∗‡, Nikolay Zamarashkin∗§
∗Institute of Numerical Mathematics of RAS, Moscow, Russia†[email protected], ‡[email protected], §[email protected]
Low tensor-rank matrices play increasingly important role in design of fast algorithms for dealingwith huge dense unstructured matrices. It is a common belief that such matrices can be the main toolin breaking “the curse of dimensionality”. The approximation techniques by low-tensor rank matricesare now well-developed. However, the inversion of such matrices is treated only in the iterative way,by using matrix multiplication and recompression. Explicit formulaes for the inversion of low-tensorrank matrices are available only in the simplest case, when the matrix has tensor rank 1:
(A⊗B)−1 = A−1 ⊗B−1.
Recently we have discovered the new class of low-tensor rank matrices that is closed under inversion,i.e. the tensor rank of the inverse matrices is bounded from above independent from the dimension.One illuminating example is the matrix of the following form:
A = I +D ⊗R+R⊗D,
where D is a diagonal matrix and R is a matrix of rank 1. Then the tensor rank of A−1 is not higherthan 5. The generalization of the result is available. The theorem comes along with a constructivealgorithm for the inversion with the required number of operations of order O(
√N), yielding a
sublinear algorithm. We will also show that almost every practically important two-level matrices(i.e., obtained from two-dimensional equation) can be reduced to the proposed form, thus leadingto a theoretically justified sublinear algorithm for the inversion of such matrices.
Fast multiplication of matrix polynomials and some tricks for the computation of thetrilinear decomposition in GF (2)
Ivan OseledetsInstitute of Numerical Mathematics of RAS, Moscow, Russia
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Suppose we want to compute the product of two matrix polynomials
C(x) = A(x)B(x), A(x) =
n−1∑
i=0
Aixi, B(x) =
n−1∑
i=0
Bixi,
where Ai and Bi are matrices over the field with two elements 0, 1 of appropriate sizes. Of course,we can compute such product as a product of two matrices with polynomial coefficients and usingwell-established algorithms for the multiplication of large integers. If Ai are, for example, 64×64 it isconvenient to store them as 64 words. Thus we are interested in some other algorithm. One possibleway is to use Karatsuba-like algorithms, but they do not achieve linear complexity. However, theirimportant property is that they use only addition and multiplication of “constants”, which in ourcase, are matrices (thus multiplication is much more computationally expensive than addition). Itwas shown in the seminal work of Kaltofen and Cantor in 1991 that polynomials can be multipliedin O(n log n log log n) time for an arbitrary algebra (however, we are not aware about its practicalimplementations). So, our main goal is to construct the algorithm that is “optimal” in some sense.All of the algorithms for computing bilinear form c = c(a, b) can be written in the following form:
c = V ((Ua) (Ub)) ,
where, for the multiplication of polynomials, a, b are vectors of polynomial coefficients, c is a vectorof coefficients for the product. The matrices U and V are independent from a, b and are (2n− 1)× rand n × r respectively. Such representation is valid not only for the computation of polynomialproduct but, for example, for the computation of a periodic convolution. We will show that r is atensor rank of a certain 2n×n×n tensor and computing U and V is equivalent to the computationof trilinear decomposition of a certain tensor. The computation of such decomposition in GF (2)is not well studied. We will present several new (sometimes heuristic sometimes not) greedy-typeapproaches that allow us to construct trilinear decomposition for such tensors, obtaining some knownas well as some new values for r. For example a multiplication of a 7 × 7 circulant by a vector inGF (2) requires only 13 “active” multiplications.
Additive Preconditioning and Aggregation in Matrix Computations
Victor PanDepartment of Mathematics and Computer Science,
Lehman College of the City University of New York, Bronx, NY 10468 USAv y [email protected]
Multiplicative preconditioning is a popular SVD-based technique for the solution of linear sys-tems of equations, but our SVD-free additive preconditioners are more readily available and betterpreserve matrix structure. We combine additive preconditioning with aggregation and other rele-vant techniques to facilitate the solution of linear systems of equations and some other fundamentalmatrix computations. Our analysis and experiments show the power of our approach, guide us inselecting most effective policies of preconditioning and aggregation, and provide some new insightsinto these and related subjects of matrix computations.
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Mappings preserving the idempotency of products of operators
Tatjana PetekUniversity of Maribor, FERI, Smetanova 17, SI-2000 Maribor, Slovenia
We will discuss mappings on the algebra of all bounded linear operators on a finite or infinitedimensional Banach space X, which preserve the idempotency of products or triple Jordan prod-ucts (⋆) of operators. The motivation for this study was the recently published paper on mappingspreserving the nilpotency of products of operators by Li, Sze and Semrl. When studying preservingidempotency instead it turns out that these mappings if unital are basically mappings on idempo-tents. About the mappings on idempotents several results are known. We use some of them in orderto obtain the characterization of all surjective unital mappings φ on B (X) having the property A⋆Bis a nonzero idempotent if and only if φ (A) ⋆ φ (B) is a nonzero idempotent (A,B in B(X)). Notethat there is no additivity assumption.
Laguerre-Pseudospectral Discretization vs. Finite-Differences for Discrete GreenFunction Calculation
Podgornova O.V., Sofronov I.L.Keldysh Institute of Applied Mathematics RAS, Moscow
The construction of non-reflecting boundary conditions for hyperbolic equations in anisotropicmedia is discussed here. We develop the idea of discrete transparent boundary conditions (DTBC)[1]. According to this approach, generation of the boundary operator is made using Green’s functionof an exterior discrete initial-boundary value problem correspondent to a discrete counterpart of theoriginal governing equations (e.g. finite-difference equations as in [1]).
We consider the circular boundary for 2D wave equation in polar coordinates for two applications:moving media and two-layer media [2]. A second-order finite-difference scheme is used for discretiza-tion here. This leads to some issues because of oscillatory nature of the discrete Green’s function.In particular, accuracy of our DTBC can be insufficient while restricting admissible computationalcosts of the boundary operator.
Trying to move ahead we looked at the other way to solve exterior problems with the requirementsof high resolution and good performance. We apply Laguerre functions expansion method for timedependence of the solution [3,4]. It results in an elliptic equation for each coefficient at Laguerrefunctions, spectrum of which has negative real part. An additional advantage is that the resolutionmatrix for elliptic equations does not depend on the order of Laguerre functions.
Further improvement of the Laguerre approach is a combination of it with pseudospectral spatialdiscretization. At this stage we consider sequence of 1D wave problems in (t, r) variables correspon-dent to 2D isotropic wave problem after azimuth Fourier decomposition. We compare efficiency ofsuch hybrid Laguerre-pseudospectral method with previous finite-difference approach using severalrepresentative test problems.
References
66
1. Ryaben’kii V.S., Exact Transfer of Difference Boundary Conditions, Functional Anal. Appl.24, (1990), 251–253.
2. Sofronov I.L., Podgornova O.V., A spectral approach for generating non-local boundaryconditions for external wave problems in anisotropic media, Journal of Scientific Computing, V. 27,3, 2006, pp. 419–430.
3. Konyukh G.V., Mikhailenko B.G., Mikhailov A.A., Application of the integral Laguerretransforms for forward seismic modeling, J. of Computational Acoustic. Vol. 9, 4, 2001, pp. 1523–1541.
4. Young-Seek Chung, Tapan K. Sarkar, Baek Ho Jung, and Magdalena Salazar-
Palma, An Unconditionally Stable Scheme for the Finite-Difference Time-Domain Method, IEEETransactions on Microwave Theory and Techniques, Vol. 51, 3, March 2003.
Exploiting rank structure in the solution of a class of nonsymmetricalgebraic Riccati equations
F. Poloni(joint work with D. A. Bini, B. Iannazzo, B. Meini)
Scuola Normale Superiore, [email protected]
Nonsymmetric algebraic Riccati equations (NAREs) XCX+B−AX−XD = 0 with structuredn× n block coefficients A,B,C,D, appear in many applications. In particular, in certain problemsof transport theory the matrices A,B,C and D are such that M =
[D −CB −A
]is a diagonal plus rank-1
M -matrix.We show how to develop structure-preserving versions of some classical algorithms for solving
NAREs in order to exploit the rank-structure of this problem. Three methods are analyzed: Newton’siteration, the structured doubling algorithm, and cyclic reduction. New algorithms are provided forimplementing the three methods above with a cost of O(n2) arithmetic operations per step instead ofO(n3), thus substantially accelerating the solution of the equation. The shift acceleration techniquefor speeding up the convergence in the critical case is implemented as well.
Reduction of matrices over a domain of principal ideals to the Smith normal formsby means of the same one-sided transformations
V. M. ProkipInstitute for Applied Problems of Mechanics and Mathematics; 3B Naukova Str. Lviv-60, 79060,
Let R be a domain of principal ideals (a commutative ring with identity e 6= 0 without nonzerodivisors of zero and such that every ideal is principal) and Rn the ring of (n × n)-matrices over R.
67
It is known that any matrix A ∈ Rn, rankA = r, is equivalent to the diagonal matrix
SA = UAV = diag(a1, . . . , ar−1, ar, 0, . . . , 0),
where U, V ∈ GL(n,R) and aj for 1 ≤ j ≤ r are the nonzero elements of R such that a1 | a2 | . . . | ar
(divides). The diagonal matrix SA is known as the Smith normal form of the matrix A (see:M. New-man. Integral Matrices. – Academic Press: New York, 1972.).
Theorem. Let A1, A2, . . . , Ak ∈ Rn be the nonsingular matrices. If (detAi,detAj) = e for alli 6= j , i, j = 1, 2, . . . , k, then there exist matrices U ∈ GL(n,R) and V1, V2, . . . , Vk ∈ GL(n,R) suchthat
UAkVk = SAk, k = 1, 2, . . . , k.
Note: The Theorem presented above is true for an adequate ring (see: O. Helmer. The elementarydivisors for certain rings without chain condition. Bull. Amer. Math. Soc., 49, 225–236 (1943)).Furthermore, the obtained Theorem can be easily extended to the elementary divisor domains R,where, for any three elements a 6= 0, b 6= 0 and c from R , there exists an element r ∈ R such thatthe greatest common divisor of (a, b, c) coincides with the the greatest common divisor of (a+rc, rb).
Bounded semigroups of matricesand the graph tractability problem
Vladimir ProtasovDepartment of Mechanics and Mathematics, Moscow State University, 119992, Moscow, Russia
This is a joint work with Vincent Blondel and Raphael Jungers (University Catolique de Louvain,Belgium).
For a given family of n × n - matrices A = A1, . . . , Am with nonnegative integer entrieswe consider the asymptotic growth of the value Fk = max‖A‖, A ∈ Sk as k → ∞, whereSk =
Ad1 · · ·Adk
, dj ∈ 1, . . . ,m
and ‖ · ‖ is an arbitrary matrix norm. Thus, Fk is the largestnorm of products of these matrices of length k. The problem is to determine whether Fk is uniformlybounded over k ∈ N, or it grows polynomially in k, or exponentially.
For general sets of matrices this problem is well-known and has found numerous applicationsin functional analysis, optimal control, approximation theory, probability etc. The rate of growthof Fk can be expressed in terms of the joint spectral radius (JSR) of the family A. However, thecomputing or estimating JSR is known to be hard even for small dimensions n. For example, formatrices with rational coefficients this problem is NP-hard, and the problem to determine whetherJSR is less than 1 is, moreover, algorithmically undecidable. We show that for nonnegative integermatrices there is a relatively simple solution and efficient polynomial time algorithms to decide thetype of growth of Fk.Theorem 1. For any family A of nonnegative integer matrices there is the following alternative:
1) A is mortal, i.e., Fk = 0 for all sufficiently large k;2) A is not mortal, but product bounded, i.e. Fk ≤ C for all k;3) A has polynomial growth, i.e. Fk ≍ k r for some natural r ≤ n− 1;
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4) A has exponential growth, i.e. Fk ≥ Cλk for some λ > 1.
Each of the conditions 1− 4 can be checked by the following criteria. We denote by S = ∪k∈NSk
the semigroup of matrices generated by A1, . . . , Am.Theorem 2. For a given family A condition (1) can be checked within O(mn2) arithmetic opera-tions. Condition (4) is satisfied iff there is A ∈ S such that Aii ≥ 2 for some i. This can be checkedwithin O(mn2) operations. Condition (3) is satisfied iff there is A ∈ S such that Aii, Aij , Ajj ≥ 1for some i, j. This can be checked within O(n3(m+ n2)) operations.
Thus, all the conditions of Theorem 1 can be decided in polynomial time. Moreover, the rate ofpolynomial growth r (condition 3) can also be found within O(n3(m+ n2)) operations.Corollary 1. For a given family A of nonnegative integer matrices the finiteness of the generatedsemigroup S can be decided in polynomial time.
These results are applied to the graph tractability problem in large sensor networks [2]. Adirected graph, whose edges are colored with m colors is called tractable if the total number of pathscorresponding to any sequence of colors is uniformly bounded by some constant.Corollary 2. The tractability of any colored graph is polynomially decidable.
References1. R. Jungers, V. Protasov and V. Blondel, Efficient algorithms for deciding the type of
growth of products of integer matrices, to appear in Linear Algebra and Applications.2. V. Grespi, G.V. Cybenko and G. Jiang, The theory of tractability with applications to
sensor networks, Technical report TR2005-555, Dartmouth College, Computer Science, Hanover NH(2005).
Singular values for the finite sections of non-Fredholm Toeplitz operators
A. RogozhinIBM (Moscow, Russia)[email protected]
Let L∞(T) denote the usual Lebesgue space on the unit circle T = t ∈ C : |t| = 1. A functiona ∈ L∞(T) is called piecewise continuous if it has one-sided limits a(t±0) for all t ∈ T. The followingcriteria for the Fredholmness of Toeplitz operators with piecewise continuous symbols is well-known.
Theorem 1. Let a ∈ L∞(T) be a piecewise continuous function. The Toeplitz operator T (a) ∈L(ℓ2) is Fredholm if and only if
a♯(t, µ) 6= 0, (t, µ) ∈ T× [0, 1],
where the function a♯(t, µ) is defined on the cylinder T× [0, 1] by
a♯(t, µ) = a(t− 0) + (a(t+ 0)− a(t− 0))µ.
In this talk we consider the behaviour of the singular values for finite sections of non-FredholmToeplitz operators with piecewise continuous symbols in the case where the symbol of the Toeplitzoperator does not vanish on the unit circle. That is we analyse the case where a♯(t, µ) = 0 for some(t, µ) ∈ T× (0, 1) ((t, µ) ∈ the interior of the cylinder).
69
A special kind of this case was considered in the papers [2,3]. In these papers the behaviour ofthe singular values for the Cauchy-Toeplitz matrices
An =
(1
i− j + γ
)n
i,j=1
, γ ∈ R,
is investigated. Note that An = Tn(ψγ) with the piecewice continuous symbol
ψγ
(eiθ
)=
π
sin γπeiγπe−iγθ, 0 ≤ θ < 2π.
It was shown that the singular values of Tn(ψγ) are clustered at π/| sin πγ|, only o(n) singular valueslie outside a given ε-neighborhood of the clustering point.
Our consideration is based on the fact that the singular values are not only the nonnegativesquare roots of the eigenvalues of T ∗
n(a)Tn(a), but also they are the approximation numbers. In viewof this fact we use the results of the paper [1]. More precisely, our investigation is leaned on thefollowing theorem (see [1], Theorem 1).
Theorem 2. Let a be a piecewise continuous function on the unit circle, and let 1 < p < ∞. Ifthe two Toeplitz operators T (a) and T (a), a(t) = a(1/t), are Fredholm in the Banach space L(ℓp),then there is an integer k ≥ 0 such that
limn→∞
sk(Tn(a)) = 0 and lim infn→∞
sk+1(Tn(a)) > 0,
where sm(Tn(a)),m = 0, 1, . . . , n, is the m−th approximation number of Tn(a), i.e. the distance ofTn(a) to the n× n matrices of rank at most n−m.
The following theorem is the main result of the talk (we order the singular values so that 0 ≤s1(Tn(a)) ≤ s2(Tn(a)) ≤ . . . ≤ sn(Tn(a)) = ‖Tn(a)‖L(ℓ2n) ).
Theorem 3. Let a be a piecewise continuous function on the unit circle such that a(t± 0) 6= 0 forall t ∈ T. If the Toeplitz operator T (a) ∈ L(ℓ2) is not Fredholm, then for all m ∈ N
sm(Tn(a))→ 0 as n→∞,
but there is an integer k ≥ 0 such that for any α > 0
sk(Tn(a)) ≥ c(α)n−α, ∀n ∈ N,
where the constant c(α) > 0 does not depend on n.
References1. A. Rogozhin, B. Silbermann, Approximation numbers for the finite sections of Toeplitz
operators with piecewise continuous symbols, Journal of Functional Analysis, 237(1), 2006, pp. 135–149.
2. E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, LAA, 149, 1991, pp. 1–18.
3. E. Tyrtyshnikov, Singular values of Cauchy-Toeplitz matrices, LAA, 161, 1992, pp. 99–116.
70
Matrix tricks in tensor approximation: 3D-cross method and maximum-volumeprinciple
Dmitry V. SavostyanovInstitute of Numerical Mathematics of RAS
We consider Tucker-like approximations with an r×r×r core tensor for three-dimensional n×n×narrays in the case of r ≪ n and possibly very large n (up to 104 — 106). As the approximationcontains only O(rn + r3) parameters, it is natural to ask if it can be computed using only a smallamount of entries of the given array. A similar question for matrices (two-dimensional tensors) wasasked and positively answered 10 years ago in [1]. We extend the positive answer to the case ofthree-dimensional tensors. More specifically, it is shown that if the tensor admits a good Tuckerapproximation for some (small) rank r, then this approximation can be computed using only O(nr)entries with O(nr3) complexity.
But the existence is good, but the algorithm is better. We also propose an algorithm to computesuch approximation, based on the same ideas as incomplete cross approximation method, proposedfor matrices in [2]. It successively computes rows, columns and “fibers” of 3D array A = [aijk](and hence is named “3D-cross method”, or simply “Cross-3D”) using a neat adaptive technique.However, the good implementation of the method (which is even better than algorithm) requires tosolve a number of matrix problems. We have found a bunch of them, rather interesting and new.For example, given a factors low-rank matrix A = UV T (A is n × n and U, V are n × r) how tofind maximum modulus element of A in O(nra) flops? (and how to provide the least a?) This,and some other questions are answered using the maximum-volume principle [3]. We are happy tonotice, that the implementation of maximum-volume principle in 3D-cross method boosts its speedup to hundreds times.
Finally, we demonstrate an application of our methods to the solution of 3D volume integralequation. For grid with 512× 512× 512 elements, the collocation matrix requires 128PB of memoryto be stored and more than a year to be computed. 3D-Cross computes the approximation withrelative accuracy 10−5 at 30 minutes and stores it in 114MB of memory.
References1. S. A. Goreinov, E. E. Tyrtyshnikov, N. L. Zamarashkin, A theory of pseudoskeleton
approximations, Linear Algebra Appl. 261 (1997), 1–21.2. E. E. Tyrtyshnikov, Incomplete cross approximation in the mosaic-skeleton method, Com-
puting, 4 (2000), 367–380.3. S. A. Goreinov, E. E. Tyrtyshnikov, The maximal-volume concept in approximation by
low-rank matrices, Contemporary Mathematics, 208 (2001), 47–51.
Spectral theory of reducible nonnegative matrices in classical and maxlinear algebra: A comparison
Hans SchneiderMathematics Department, University of Wisconsin
71
We consider two forms of nonnegative linear algebra: classical and max. We compare the spectraltheories of reducible nonnegative matrices in the two forms of linear algebra, stressing the similaritiesand identifying the differences.
Exponential parametrization and linear scaling Density Functional Theory inBigDFT
Reinhold SchneiderUniversity of Kiel
Electronic structure calculation plays an increasing role in the numerical simulation of molecularphenomena. Density functional theory povides an effective one particle model for the computationof the groundstate energy of a quantum mechanical system of N electrons moving in a Coulombicfield given by fixed nuclei. Instead of the high dimensional Schrodinger equation o one has to solvenonquadratic optimization problem in three space dimensions.
Since the exchange correlation potential is not yet known there remains a modelling error. Thegroundstate energy is the minimum of the Kohn-Sham engergy functional depending on N pairwiseorthogonal orbital functions. This is a minimization problem on a prescribed Stiefel resp. Grassmanmanifold. This problem can be treated either as a constraint optimization problem, or by localparameterization of the manifold. The analysis of gradient type minimization procedure in bothcases will be considered, and the connection of both approaches will be highlighted.
For insulating materials it is known, that the subspace generated by the orbital functions canbe spanned, approximatively, by local functions, e.g. Wannier orbital. Linear scaling with respectto the number of atoms of the computational costs can be achieved by enforcing the locality of theiterates in each iteration step. For this purpose local, or better multi-scale high order basis functionsin conjunction with smooth, nonlocal, pseudo-potentials can provide an efficent alternative to eitherplane wave basis sets or Gaussian functions. This approach has been integrated in the BigDFTelectronic structure program.
The matrix of bordering functions and the problems of the restoration of signals ondiscrete samples
A. V. SedovThe Southern Scientific Centre of Russian Academy of Sciences, 132, Prosveshchenia Str.,
Novocherkassk, Rostov Region, 346428, Russiasedov [email protected]
The classical theorem of sampling (Kotelnikov-Shannon’s theorem) for an infinite signal x(t),−∞ < t <∞ in time, with the final energy and the spectrum x(ω) limited by frequency ωh, defines
72
the condition T < π/ωh of formation of discrete samples x [kT ] = x (kT ) , k ∈ Z and the formula ofthe exact restoration of a signal on these samples:
x(t) =∞∑
k=−∞x[kT ] sinc( t
T − k) =(S
t/T∞
)TX∞
p , (1)
where X∞p = [..., x[2T ], x[T ], x[0], x[−T ], ...]T is the infinite-dimensional vector of samples x[kT ]
; St/T∞ =
[. . . , S
t/T−2∞ , S
t/T−1∞ , S
t/T∞ , S
t/T+1∞ , . . .
]Tis the infinite-dimensional vector consisting of
equidistant samples of a function St/T−k∞ = sinc (t/T − k).
In case of a final signal xp(t) , t ∈ [0, TS) in time which can be considered as the period of thesignal x(t) with the final energy and with the spectrum x(ω) limited by frequency ωh the otherformula of restoration [1]
x(t) =N−1∑k=0
xp[kT ]
[1N + 2
N
<N/2>∑n=1
cos(2πn (t/T−k)
N
)]=
N−1∑k=0
xp[kT ]St/T−kN (2)
on discrete samplesxp[kT ], k = 0, N − 1
is offered, where T = TS/N and the condition T < π/ωh
is satisfied. The bordering function St/T−kN is a function of samples for a final signal xp(t) , that
is it is the same for each of N samples xp[kT ], but it is displaced on k steps on an axis of time.Besides it is the periodic one with the period N . The sign 〈N/2〉 is the integer part of number N/2.
The interrelation of formulas (1) and (2) and so the functions of samples sinc( tT − k) and S
t/T−kN is
defined by the limit limN→∞
St/T−kN = sinc (t/T − k).
In practical tasks of interpolation the formula (2) is used for the reception on samples xp[kT ] ,k = 0, N − 1 of the signal xp(t) of the greater number of samples xp[kT
′] , k = 0,m− 1,(m≫ N)with the small period of sample T ′ = (Tc/m). Thus the formula (2) can be written as:
Xmp = Sm×N XN
p =[Sm,0
N , Sm,1N , . . . , Sm,N−1
N
]XN
p (3)
where Xmp = [xp[0], xp[T
′], xp[2T′], ..., xp[(m− 1)T ′]]T is the m-dimensional vector; XN
p = [xp[0],
xp[T ], xp[2T ], ..., xp[(N − 1)T ]]T is the N -dimensional vector; Sm×N is the matrix of bordering func-tions. The columns of matrix Sm×N are the vectors:
Sm,kN =
[S−k
N , SN
mT−k
N , S2NmT
−k
N , ..., SN(m−1)
mT−k
N
]T
, Sm,kN ∈ Rm.
According to the expression (3) the formula of the restoration ( the interpolation) of a final signalxp(t) in time, with properties mentioned above, can be submitted as the product of the matrix Sm×N
of bordering functions on the vector XNp of the samples measured on conditions that T < π/ωh. The
expression (3) can be considered, as a new interpolation formula as a kind of the decomposition of
the signal (the function) xp(t) by columns Sm,kN of the matrix Sm×N . Its feature is that the samples
xp[kT ], k = 0, N − 1 of the signal xp(t) are the factors of decomposition, so their calculation is notrequired.
References1. A. V. Sedov, The specification of the sampling theorem and the formula of the restoration
of a signal on discrete samples, Electromechanics, 2001. Vol. 2, pp. 52–59.
73
The property of the quasi-orthogonality of vectors and the quasi-unitary matrixesat the modelling of signals
A. V. SedovThe Southern Scientific Centre of Russian Academy of Sciences, 132, Prosveshchenia Str.,
Novocherkassk, Rostov Region, 346428, Russiasedov [email protected]
We shall stop on some properties of the matrix of bordering functions Sm×N (see the previousreport). The particular properties of this matrix allow to receive a number of the general propertiesand definitions, namely, the property of the quasi-orthogonality of vectors and the definition of thequasi-unitary matrix.
1.The property of the quasi-orthogonality of vectors.We shall construct Gram’s matrix GN×N for the vectors Sm,k
N of the matrix Sm×N of borderingfunctions:
GN×N =(Sm×N
)TSm×N =
s1I
N×N , forN− odd;
s1IN×N − s2TN×N = EN×N , forN− even,
where IN×N is an unitary matrix of dimension N × N ; TN×N =[
(−1)i ·j]
i,j=1,Nis Toeplitz’s
matrix, it elements are equal along each diagonal; s1 = m/N ; s2 = (1− (2 〈N/2〉+ 1)/N)m/N .
In case of the odd value N the matrix GN×N is the diagonal one, and the vectors Sm,kN are
the orthogonal ones. In case of the even value N the matrix GN×N is not the diagonal one, andthe vectors Sm,k
N are not the orthogonal ones. However, if in the latter case the samples xp[kT ],k = 0, N − 1 of the signal xp(t) are chosen from the condition T < π/ωh, and it is easy to show, that
the formula Xmp = Sm×N XN
p also is true, as in case of the orthogonal vectors Sm,kN . We shall define
the given property as the quasi-orthogonality vectors Sm,kN of the matrix Sm×N or the harmonious
quasi-orthogonality.2. The definition of the quasi-unitary matrixes.The quasi-unitary matrix for the square matrix XN×N
p of dimension N ×N , where N is an even
value, which we shall name a matrix EN×N = N/m EN×N such as
XN×Np = EN×NXN×N
p . (1)
The distinctive feature of the quasi-unitary matrixes EN×N is that it is not a diagonal matrixIN×N with units on a diagonal, but at the same time it possesses its basic property (1) for thematrix XN×N
p . Thus any columns of the matrix XN×Np are vectors XN
pl , l = 0,N − 1 consisting of
the samples xpl[kT ], k = 0, N − 1 of different signals xpl(t) with the spectra limited by frequencyωh, and the condition T < π/ωh is satisfied for the sampling interval. The expression (1) is true and
for the columns XNpl = EN×NXN
pl . The inverse matrix(EN×N
)−1also is the quasi-unitary matrix
for the matrix XN×Np .
The equality (1) is broken, if in the spectrum of the signals xpl(t) making the columns XNpl of the
matrix XN×Np , there is a harmonic with number M = N/2 , and this harmonic is not a sine with a
zero phase.It is possible to sum up that we can find for the matrixes of a kind XN×N
p besides the unitary
matrix IN×N not a diagonal matrix for which the condition (1) is also satisfied.
References
74
1. A. V. Sedov, The specification of the sampling theorem and the formula of the restorationof a signal on discrete samples, Electromechanics, 2001. Vol. 2, pp. 52–59.
Automorphisms of the semigroup of invertible matrices with nonnegative elementsover commutative partially ordered rings
Bunina E. I., Semenov P. P.Department of Mechanics and Mathematics, Moscow State University, 119992, Moscow, Russia
In the paper [1] A. V. Mikhalev and M.A. Shatalova described all automorphisms of the semi-group Gn(R) of invertible matrices with nonnegative elements in the case when R is a skewfield andn ≥ 2. In [2] E. I. Bunina and A.V. Mikhalev described all automorphisms of the semigroup Gn(R),if R is an arbitrary linearly ordered associative ring with 1/2, n ≥ 3. In this paper we describeautomorphisms of the semigroup Gn(R) in the case when R is a commutative partially ordered ringwith 1/2, 1/3 ≥ 0, n ≥ 3.
Let Σn be the symmetric group of the order n, Sσ be the matrix of transposition σ ∈ Σn,Sn = Sσ|σ ∈ Σn. By Dn(R) we denote the group of all invertible diagonal matrices from Gn(R).By Bij(x) we denote the matrix E + xeij . Let P be the subsemigroup in Gn(R), generated by allmatrices Sσ (σ ∈ Σn), Bij(x) (x ∈ R+, i 6= j) and Dn(R).
Two matrices A,B ∈ Gn(R) are called P-equivalent (see [1]), if there exist matrices Aj ∈ Gn(R),
j = 0, . . . , k, A = A0, B = Ak, and matrices Pi, Pi, Qi, Qi ∈ P, i = 0, . . . , k − 1 such that PiAiPi =QiAi+1Qi. By GE+
n (R) we denote the subsemigroup in Gn(R), generated by all matrices, that areP-equivalent to the matrices from P.
For every matrix M ∈ Γn(R) let ΦM denote an automorphism of Gn(R) such that ∀X ∈ Gn(R)ΦM(X) = MXM−1.
For every y(·) ∈ Aut(R+) by Φy we denote an automorphism of Gn(R) such that ∀X = (xij) ∈Gn(R) Φy(X) = Φy((xij)) = (y(xij)).
Theorem. Let Φ be an automorphism of the semigroup Gn(R), n ≥ 3, 1/2, 1/3 ∈ R+. Thenon the semigroup GE+
n (R) Φ = ΦMΦcΩ, where M ∈ Γn(R), c(·) ∈ Aut(R+), Ω(·) is a centralhomotethy of GE+
n (R).
References1. A.V. Mikhalev, M.A. Shatalova, Automorphisms and antiautomorphisms of the semi-
group of invertible matrices with nonnegative elements, Math. Sbornik, 1970, 81(4), 600–609.2. E. I. Bunina, A.V. Mikhalev, Automorphisms of the semigroup of invertible matrices with
nonnegative elements, Fundamental’naya i prikladnaya matematika, 2005, 11(2), 3–23.
On extensions of the closure operation for matrices over the max-plus semiring
Sergeı Sergeev
75
Department of Physics, Sub-Department of Quantum Statistics and Field Theory, Moscow StateUniversity, Moscow, 119992 Leninskie Gory, Russia
The research was supported by the RFBR grant 05-01-00824 and the joint RFBR/CNRS grant05-01-02807.
Algebraic closure of a square matrix A over the max-plus semiring Rmax = R ∪ −∞ is theseries I ⊕ A ⊕ A2 ⊕ . . ., where I is the identity matrix. This series converges iff λ(A) ≤ 1, whereλ(A) is the maximal cycle mean of A.
I show that the closure operation can be extended, by some kind of renormalization, over themax-plus matrices which have at least one finite diagonal (whose weight is not equal to −∞). Thishas been published in [2]. I also show that further extensions can be obtained by making use of thecellular decomposition and other concepts considered by Develin and Sturmfels [1].
References1. M. Develin and B. Sturmfels, Tropical convexity, Documenta Math., 9:1–27, 2004. Also
arXiv:math.MG/0308254.2. S. Sergeev, Max-plus definite matrix closures and their eigenspaces, Linear Algebra Appl.,
421:182–201, 2007. E-print arXiv:math.MG/0506177.
Tridiagonal canonical matrices of bilinear and sesquilinear forms
Vladimir V. SergeichukInstitute of Mathematics, Tereshchenkivska 3, Kiev, Ukraine
This is joint work with Vyacheslav Futorny and Roger A. Horn.We give tridiagonal canonical forms of matrices of
(i) bilinear forms and sesquilinear forms,
(ii) pairs of forms, in which each form is either symmetric or skew-symmetric, and
(iii) pairs of Hermitian forms
over an algebraically closed field of characteristic different from 2. Our canonical forms are directsums of matrices or pairs of matrices of the form
ε a 0c 0 b
d 0 ac 0 b
d 0. . .
0. . .
. . .
;
they employ relatively few different types of canonical direct summands.
76
References1. V.Futorny, R.A.Horn, V.V. Sergeichuk, Tridiagonal canonical matrices of bilinear
or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms, PreprintRT-MAT 2006-16, Universidade de Sao Paulo, 2006. Submitted for publication.
The computing of the stationary magnetic field in the presence of surface withinfinite permeability
K. S. ShaposhnikovSouthern Scientific Centre of Russian Academy of Science
cyr @mail.ru
We consider the problem of stationary magnetic field computation in presence of piecewise-smooth Lipschitz surface Γ′ with infinite permeability. The media is uniform and isotropic, i.e.µ = const.
The problem of magnetic field computing in classic formulation can be reduced to the followingboundary problem:
∆ϕ∗ = 0 outside Γ′;ϕ∗ = C − ϕ0 on Γ′;ϕ∗ (M) −→M→∞
0, (1)
where ϕ∗ is scalar potential of intensity of reaction field H∗, ϕ0 is a potential of non-perturbed fieldintensity H0, H = H∗ + H0, C = const.
We use the orthogonal projections method for the problem (1) solving [1]. We denote Γ′′ is theclosure of Γ′ to the closed surface Γ = Γ′ ∪Γ′′∪ ∂Γ, ∂Γ is the adjacent boundary between Γ′ and Γ′′.Ω− is space portion, bounded by Γ, Ω+ – its outside, Ω = Ω− ∪ Ω+ ∪ Γ. We orient Γ an outwardnormal with respect to Ω−.
We consider a Hilbert space H (Ω) of the real functions that is introduced and studied in [2].The elements of H (Ω) have a finite Dirichlet integral, they are harmonic out of ball ΩR : Ω− ⊂ ΩR,and vanish at infinity [2]. Inner product and norm in H (Ω) are following:
(ψ1, ψ2)H =
∫
Ω
∇ψ1∇ψ2dΩ, ‖ψ‖H =√
(ψ,ψ)H.
According [2], there is the orthogonal decomposition for this space: H (Ω) = H∗ (Ω) ⊕ H0σ (Ω).
The subspace H∗ (Ω) consists of the functions with constant value on Γ. H0σ (Ω) consists of the
functions that representable by potentials of simple layer with the zero mean of densities on Γ.We choose the potentials of H0
σ (Ω) that have zero densities on Γ′′. The set of such functions
form a subspace in H0σ (Ω) [1] that is denoted as H
0(1)σ (Ω). We have shown that the orthogonal
complement for its subspace in H0σ (Ω) is the subspace that consists of potentials with the constant
values on Γ′ with the zero mean of densities on Γ. We denote this subspace as H0(2)σ (Ω).
We shall interpret ϕ0 as the potential of equivalent non-perturbed field that is potentially in Ωand has the finite energy in Ω. The equivalence is understood in sence of [3]. This mean that ofpotentials of the initial and the equivalent fields have coincidetnd values on Γ′. We have shown that
if ϕ0 is in the H (Ω) then ϕ∗ belongs to H0(1)σ (Ω) and the potential of the equivalent resulting field
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ϕ = ϕ0+ϕ∗ is in H0(2)σ (Ω). Thus, the solution of the problem is reduced to ϕ0 projection to subspace
H0(1)σ (Ω) that can be found by the calculation of ϕ0 coordinates cknk=1 in some basis ϕknk=1 for
H0(1)σ (Ω). Then ϕ∗ is represented by the following: ϕ∗ ∼=
n∑k=1
ckϕk. Coordinates are determined by
the solution of the system of linear algebraic equations with the Gram’s matrix in H (Ω):
(ϕ1, ϕ1)H . . . (ϕ1, ϕn)H. . . . . . . . .
(ϕn, ϕ1)H . . . (ϕn, ϕn)H
c1. . .cn
= −
(ϕ0, ϕ1
)H
. . .(ϕ0, ϕn
)H
. (2)
We use coordinate functions that are represented by potentials of simple layer with finite piecewiseconstant densities on Γ′. Densities is zero on Γ′′. We have proved that sush system of functions is
complete in H0(1)σ (Ω) in sence of Ritzs’s approximate sequence convergence. In this case the elements
of SLAE (2) are defined by the following formulas:
(ϕi, ϕk)H =1
4π
∫
Γi
σi
∫
Γk
σk
rdΓdΓ,
(ϕ0, ϕi
)H
=
∫
Γi
ϕ0σidΓ.
The computing algorithm and effective software package for its realization are developed.
References1. Astakhov V. I., Kochubey T. V., Shaposhnikov K. S., The orthogonal projections
method in tasks of stationary magnetic fields calculation, Studies of the Southern scientific centre ofthe Russian academy of sciences, Rostov-on-Don, SSC RAS Publishing house, 2007, pp. 51–72. (inRussian)
2. Astakhov V. I. Surface potentials and operators of potential’s theory in Dirichlet’s spaces,Electromechanics, 2000, 2, pp. 3–18. (in Russian).
3. Astakhov V. I., About an admissibility of idealization of borders of polarizable bodies andsome energetic identities for stationary magnetic and electrostatic fields, Electromechanics, 2000,1, p. 3–14. (in Russian).
On discrete projection methods for rotating incompressible flow with Coriolis force
Andriy Sokolov∗, Maxim A. Olshanskii† and Stefan Turek∗
∗Institute for Applied Mathematics, University of Dortmund, Germany†Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow,
Russia∗asokolow,[email protected], †[email protected]
The construction of an efficient solver for the incompressible system of Navier-Stokes equations is along-term purpose of CFD researchers. Since decades an evident progress is observed: large varietyof methods and algorithms were proposed and implemented into commercial and open-source code.
In many physical and industrial processes there is a necessity of numerical simulations of modelswith moving boundaries in a 3-dimensional space. As proposed in the literature, approaches likefictitious boundary [4] or Arbitrary Lagrangian Eulerian methods require large amount of CPU time
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to simulate even a 2D benchmark models. Moreover, they could provide a source of additional errorsin velocity and pressure fields. At the same time, there is a large class of rotating models, where theuse of the above methods can be avoided by some modifications of Navier-Stokes equations and/ortransformations of the coordinate system.
While performing numerical simulation for the Stirred Tank Reactor problem (fig. 1, left), wecan avoid moving boundary parts (blades of the rotating propeller) by transforming the coordinatesystem in such a way that the propeller is considered to be motionless, but the wall of the tankrotates. By doing so, we are required to perform a numerical simulation for the system of Navier-Stokes equations with rotational forces:
∂u
∂t + (u · ∇)u− ν u + 2ω × u + ω × (ω × r) +∇p = f
∇ · u = 0
where 2ω × u and ω × (ω × r) are the so-called Coriolis and centrifugal forces, respectively.
Figure 1: STR benchmarks. (LEFT) mesh (RIGHT) simulation
Thus, in our talk we present a numerical analysis and algorithmic details for treating the systemof Stokes and Navier-Stokes equations with Coriolis force term. Using a Discrete Projection Methodwith modified Pressure Schur Complement operator, we examine the influence of the Coriolis forceon every step of the algorithm: modification of the momentum equation and building of a new blockdiagonal preconditioner, construction of a new Pressure Schur Complement operator by insertingthe off-diagonal parts, which are due to the ”rotating” terms and an explicit formation of its inverse,and finally, correction of pressure and velocity to satisfy the incompressibility constraints. Detailednumerical studies of the improvements in the convergence rate, its dependence on the magnitude ofthe time step and angular velocity are provided.
We also consider the addition of the convection term into the Schur Complement, discuss itsinfluence on the iterative behaviour of the solver and propose a coriolis-convection oriented PressureSchur Complement.
Finally, we provide numerical simulations for polygonal geometries and check the efficiency of oursolver by comparing simulation results (fig. 1, right) with those obtained by the fictitious boundarymethod for moving boundary parts.
The used solver is based on the 3-dimensional PP3D CFD-code from the Featflow package(www.featflow.de), [2,3].
References1. M. A. Olshanskii, An iterative Solver for the Oseen Problem and Numerical Solution of
Incompressible Navier-Stokes Equations, Numer. Linear Algebra Appl., Vol. 6, page 353–378, 1999.2. S.Turek, On discrete projection methods for the incompressible Navier-Stokes equations: an
algorithmical approach, Comput. Methods Appl. Mech. Engrg., Vol. 143, page 271–288, 1997.3. S.Turek, Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Compu-
tational Approach, LNCSE 6, Springer, 1999.
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4. S. Turek, D. C. Wan and L. P. Rivkind, The Fictitious Boundary Method for theimplicit treatment of Dirichlet boundary conditions with applications to incompressible flow simu-lations, In E. Bansch editor, Challenges in Scientific Computing CISC 2002, LNCSE, page 37–68,Berlin, 2002. Springer, Berlin.
A New Method for a Quadratic Inverse Eigenvalue Problem in Structural Dynamics
Vadim Sokolov(with Biswa Datta, Sien Deng, Moody Chu and Daniil Sarkissian)
Northern Illinois University, DeKalb, USAThe research of Biswa Datta has been supported by NSF grant #DMS-0505784.
In structural dynamics, it is very often required that an analytical model be updated using theeigenvalues and eigenvectors measured from a real-life or a prototype structure. Mathematically,the problem is an inverse eigenvalue problem for the associated quadratic matrix pencil. Despitemuch research done by scientists and practicing engineers, the problem has not been satisfactorilysolved. Most of the industrial solutions are ad hoc in nature and lack solid mathematical foundation.Furthermore in practice almost always the problem is solved for the undamped model, thus theunderlying inverse problem is the one for a linear matrix pencil, which makes problem much easierto solve. Most of these methods also assume that the measured data is accurate and proceed toupdating the model under that assumption. In reality, the measured data has always some noiseand damping should not be neglected.
In this talk, we present a new optimization method for a damped finite element model updating.The method consists of two stages. In Stage I, the measured data is updated so that the updatedeigenvectors satisfy a recently established orthogonality condition involving mass and damping ma-trix. In stage II the stiffness matrix is updated so that the quadratic pencil has the prescribedeigendata.
An approach to feature-based basis construction for image analysis viaKarhunen-Loeve transform
A. Yu. SolodovschikovKeldysh Institute of Applied Mathematics RAS, Moscow
anton [email protected]
The paper is devoted to an investigation of discrete and continuous variants of the Karhunen-Loeve transform. An algorithm of analytical construction of a basis with non-correlated expansioncoefficients for analyzing digital images based on this expansion is presented. The results obtainedare compared with the existing results in the problem of edge detection. As an example, the problemof formal construction of operators similar to the Hueckel operator by this method is solved. Thecharacteristics of the formed basis functions are compared.
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Usually, natural scenes consist of a set of three-dimensional objects. TV digital or photo im-ages are two-dimensional projections of these scenes and consist of regions with different intensitycharacteristics of particular pixels. To obtain a vector description of the source digital image ofthree-dimensional scenes, the operator proposed by M. Hueckel that transforms a digital image toa set of linear segments of a fixed length can be successfully applied. In his paper, Hueckel solved amore particular problem of mapping a digital intensity image in a circular window of a fixed radiusinto the equation of the straight line that approximates the position and direction of the line of max-imums of the intensity gradients at points of the window. For this purpose, he proposed a totalityof smooth basis functions and their discrete representation for a two-dimensional expansion withinthe window. The detailed investigation of these functions has shown that they satisfy the criterionof the minimum error between the elements of the source image and the most probable position ofthe line of intensity gradients. The important idea of the statement of the problem of analyzing animage that states that it is necessary to find a convenient basis of a certain integral transform that isadequate to the problem solved is contained in. This approach forestalls the later method of wavelettransforms, which, currently, have noticeably pressed the two-dimensional Fourier transforms.
Indeed, to obtain a concise description of the scene content, it is reasonable to select from itsimage only “necessary” (depending on the desired content) elements of the spatial interrelations ofpixel intensities eliminating “unnecessary” (excessive) information and compensating dropped data(as a result of noise in sensors). The problem of constructing a method of image description such thatdescription elements are weakly dependent arises. One of the most promising methods for solvingthis non-simple problem is to find a two-dimensional orthogonal basis for an integral transform of adigital image that is adequate to the problem of scene recognition. It is important that this basisshould provide that the desired information is to be concentrated in the minimum number of theexpansion coefficients. As a rule, these “higher harmonics” of the expansion spectrum contain thedesired information about the integral intensity discontinuities, while the high-frequency componentsof the spectrum reflect only excessive information, e.g., in the problem of finding contours of imageregions.
The performed testing has confirmed the adequacy of the Hueckel basis to the problem posedby him. However, it is extremely desirable to obtain a formal method for constructing a basis thatis adequate to the solved problem of image processing. We have managed to obtain this methodapplying another Kahrunen-Loeve transform (KL-transform), which is well-known in the recognitiontheory. In contrast to a number of papers that develop a similar approach, the investigation of themethod is not limited to the discrete variant of the transform. The use of the continuous KL-transform allows one to describe the structure of the basis analytically, which is also convenient inthe scale analysis.
In literature, the KL-transform is described within the probabilistic signal model. However,unconditionally, the Kahrunen-Loeve transform has more general nature. It is applicable for atotality of image elements that are not mandatory random, for which there exists knowledge of ananalogue of the correlation function of their intensity. The KL-transform has a number of propertiesthat distinguish it among the other linear transforms. The main drawback of the KL-transformis that there are no fast algorithms for its computation. The method obtained in the paper isdescribed by examples of constructing operators for transforming images similar to the Hueckeloperator. A formal derivation of the basis for the Hueckel operator is presented and the significanceof its harmonics is analyzed.
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On some fundamental properties and applications of Chebyshev continued fractions
Sergey P. SuetinSteklov Mathematical Institute of RAS
The research was carried out with the financial support of the Russian Foundation for BasicResearch (grant no 05-01-01027) and the Programme of Support of Leading Scientific Schools
(grant no NSh-4466.2006.1)[email protected]
1. Let µ be a positive Borel measure with a compact support on R, function
µ(z) :=
∫dµ(x)
z − x , z /∈ suppµ, (1)
be a Markov function. In 1855 P. Chebyshev [1] assosiated to the function of type (1) the continuedfraction
µ(z) ∼ Cbµ(z) =a1
z − b1 −a2
z − b2 −· · · −
an
z − bn −· · · , (∗)
where aj > 0, bj ∈ R, a1 = µ(R). The n-th convergent Cbµn (z) = Pn(z)/Qn(z) to (∗) has the following
fundamental property
Cbµn(z) =
s0z
+s1z2
+ · · ·+ s2n−1
z2n+ · · · , sk =
∫xk dµ(x), (2)
polynomial Qn is of n degree and is orthogonal with respect to µ.2. By A. Markov theorem [2] Cbµ
n → µ, n→∞, locally uniformly in C \ suppµ.Let suppµ = [α, β], µ′ > 0 a.e. on [α, β] and rational function r ∈ C(z) ∩H[α, β]. Denote by
G = f : f = µ+ r for a segment [α, β]. Let f be given by its power series expansion at infinity
f =
∞∑
k=0
ckzk+1
. (3)
A. Gonchar [3] proved that if f ∈ G then Cfn → f in spherical metric locally uniformly in C \ [α, β].
In particularly each pole of f of multiplycity m attractes exactly m poles of Cfn . When measure µ
is complex but “smooth” enough the m poles of Cfn , f = µ+ r, asymptoticaly form the vertices of
a regular m-polygon (A. Gonchar, S. Suetin [4]). The last result is valid for the functions of typef(z) = r1(z)/
√(z − α)(z − β) + r2(z), where r1, r2 ∈ C(z).
From the Gonchar’s theorem it follows that for f = µ+ r ∈ G
f(z) = Cfn0
(z) +
∞∑
k=n0
Ak
(QkQk+1)(z), z ∈ C \ suppµ, (4)
i.e. the Chebyshev continued fraction Cf (z) provides a nonlinear method of summation of powerseries (3).
Let f be an algebraic function given by (3). Then by H. Stahl [5] theorem Cfn
cap→ f on thecompact subsets of D := C \ S, where f has the single-valued meromorphic continuation in D andthe compact S has a special “symmetric” property (as a segment on R). From Stahl’s theorem itfollows that
f(z)cap= Cf
n0(z) +
∞∑
k=n0
Ak
(QkQk+1)(z), z ∈ C \ S. (5)
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3. The n-th convergent Cfn to Chebyshev continued fraction Cf depends only on the first 2n
coefficients c0, c1, . . . , c2n−1 of (3) just as the partial sum Sf2n. But from (4) and (5) it follows that
in numerical analysis the Cfnn∈N has a great advantage before the partial sums Sf
2nn∈N.To illustrate that fact we calculated numerically [6] the limit cycle of the very well-known free
van der Pol equationU + U = εU(1− U2), U = U(t, ε),
for some values of ε, where ε is a “small” parameter, ε ∈ [0,+∞).
References1. P. Tchebycheff, Sur les fractions continues, Journ. de math. pures et appl. Ser. 2, 3
(1858), 289–323.2. A. Markoff, Deux demonstrations de la convergence de certaines fractions continues, Acta
Math., 19 (1895), 93–104. Mathscinet 1554864.3. A.A. Gonchar, On convergence of Pade approximants for some classes of meromorphic
functions, Mat. Sb., 97(139): 4(8) (1975), 607–629. Mathscinet 387552.4. A.A. Gonchar, S. P. Suetin, On the Pade approximants to Markov type meromorphic func-
tions, Sovremennye problemy matematiki (In Russian), 5, Steklov Mathematical Institute, Moscow,2004. Mathscinet 2141293.
5. H. Stahl, Orthogonal polynomials with complex valued weight function, I, II, Constr. Ap-prox., 2 (1986), 225–240; 241–251. Mathscinet 891973.
6. A.A. Gonchar, S. P. Suetin, Numerical analysis of some parameters of limit cycle of thefree van der Pol equation, in Russian, in preparation.
Asymptotic solution of large Toeplitz systems with an infinitely differentiable symbol
Sergey Sukhoverkhov, Igor B. SimonenkoSouthern Federal University, Rostov-na-Donu, Russia
We consider the asymptotics of the solutions of large linear systems with Toeplitz matricesgenerated by a complex valued symbol σ which is infinitely differentiable, has no zeroes and has azero argument increment. We construct an asymptotic inverse of n × n Toeplitz matrix T n
σ , whichapproaches exact inverse faster then any positive degree of 1/n as n tends to plus infinity. If the righthand side vector b is quasi-polynomial, i.e., b(k) = eiαk(k − k0)
m we obtain an asymptotic formulafor the solution of T n
σ x = b. Using this formula different unknowns xk could be found independentlyof each other. That is, in order to find particular components of the solution there is no necessityto solve the whole system. Special attention will be paid to the asymptotic formula of the centralpart of the solution, because it has more simple form.
Matrix approach to model of polarized radiation transfer in heterogeneous systems
83
Tamara A. Sushkevich, Sergey A. Strelkov, Sveta V. MaksakovaKeldysh Institute for Applied Mathematics, Russian Academy of Sciences 4, Miusskaya Ploschadj,
Moscow, 125047 RussiaThe work have been supported by the Russian Foundation for Basic Research (projects
06-01-00666, 05-01-00202)[email protected]
The mathematical models of the matrix-vector transfer operators are suggested to computeStoke’s parameters vector as the solution of the general boundary-value problem in polarized ra-diation transfer theory for the 1d, 2d, 3d plane layers and spherical shell’s. Space and angulardistributions of polarized radiation inside the relevant layer of these media as well as reflected andpassed through the layer radiation are formed as a result of multiple scattering and absorption. Anew approach is proposed to radiation transfer modeling in optically thick layers that are repre-sented as a heterogeneous multi-layer system each layer of which is described by different opticalcharacteristics.
The kernels of the matrix-vector transfer operators are the tensors of the influence functions orthe tensors of the spatial-frequency characteristics. The tensors of the influence functions of eachlayer are determined by solutions of the radiation transfer first boundary vector problem with anvector external source function while the contribution of multiple scattering and absorption in thelayer is taking into account (Sushkevich, 2005).
The basic models of the tensors of the influence functions and the tensors of the spatial-frequencycharacteristics are stated. The matrix-vector operation, which describe one act of polarized radiationinteraction with internal and external boundaries and take into account multiply scattering in thelayers through tensor of the influence functions, is determined.
The boundary vector problem for each layer can be solved depending on optical thickness, scat-tering and absorption characteristics by one of the following techniques: i) as a solution of transferequation with an azimuth dependence; ii) as a solution of the problem with azimuth symmetry; iii)as a solution in two-flux approach; iv) as an approximate solution in asymptotic approach. Thematrix operators of radiation transmittance and reflectance on the boundaries between the layersare formulated based on the collision integrals and the separate layers are united in a system bythese operators. To calculate the total radiation inside or on the boundaries of the system withradiation exchange between the layers a matrix-vector operation is constructed for the operators thekernels of which are given by the tensor of the influence functions of the layers.
The representation of the solution to boundary-value vector problem as a functional is the transferoperator of the radiation transfer system which establishes the explicit relationship between therecording radiation and the “scenarios” (the optical image) at the dividing boundaries of media.In turn, by the use of the tensors of the influence functions, the “scenarios” is described clearlythrough the characteristics of the reflection and transmission of the dividing boundaries at the givenits illumination. The tensors of the influence functions are invariant about the conditions of theillumination and the properties of the dividing boundaries.
This approach enables to simulate radiation fields in a wide range of variations of the opticalcharacteristics of these layers and to analyze mechanisms of radiation characteristics formation insideand outside the layers as well as to estimate any contribution of each region. Reflected radiation anddistributions of the polarized radiation characteristics inside the layer near to the border illuminatedby an external flux are thus calculated with higher accuracy than in common-used techniques.
Original mathematical tools are proposed for the first time to model radiation transfer in multi-layer non-homogeneous heterogeneous media with different radiation regimes within separate regionsof the system. The proposed approach is based on construction of generalized solutions in the formof matrix functionals, tensors of influence functions for each layer of the system serve as kernels of
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the functionals. The tensors of the influence functions of the layers with different characteristicsof scattering and absorption and radiation regimes can thus be calculated by various methods indifferent approaches of radiation transfer theory.
Development of new mathematical tools to be realized on high-output multi-processor computersystems is needed to solve direct and inverse problems of radiation transfer theory for naturalheterogeneous media as related to international cooperation on air-space global monitoring of theEarth as well as to an international global project on research.
New opportunities of the proposed model are linked with verification of engineering appliedtechniques and plain layers approach which are often used for an express-analysis of space data inremote sensing and in radiation blocks for models of climate, circulation, forecast, photo-chemicalkinetics, dynamics of ozone-sphere, trans-boundary displacement of pollutions for an air basin, etc.
References1. Sushkevich T.A., Mathematical models of radiation transfer, Moscow, BINOM. Knowledge
Laboratory, 2005. 661 p.
Spectral filtering methods with Reflective/Anti-Reflective boundary conditions
Cristina Tablino-PossioDipartimento di Matematica e Applicazioni,
Universita di Milano Bicocca, via Cozzi 53, 20125 Milano, ItalyThe work of the author is supported in part by MIUR grant no. 2006017542.
In this talk, we analyze and compare some spectral filtering methods as truncated singular/eigen-value Decompositions and Tikhonov/Re-blurring regularizations in the case of the recently proposedReflective [10] and Anti-Reflective [11] boundary conditions.
Numerical evidence is given to the fact that spectral decompositions (SDs) provide a good imagerestoration quality and in particular the Anti-Reflective spectral decomposition, despite its lossof orthogonality. The related computational cost is comparable with previously known spectraldecompositions.
The model extension to the cross-channel blurring phenomenon of color images is also consideredand the quoted spectral filtering methods are suitable adapted.
1. A. Arico, M. Donatelli, S. Serra Capizzano, Spectral analysis of the anti-reflectivealgebra, Linear Algebra Appl., to appear.
2. A. Arico, M. Donatelli, J. Nagy, S. Serra-Capizzano, The anti-reflective transformand regularization by filtering, 2006, submitted.
3. M. Bertero, P. Boccacci, Introduction to inverse problems in imaging, Inst. of PhysicsPubl. London, UK, 1998.
4. M. Donatelli, C. Estatico, A. Martinelli, S. Serra Capizzano, Improved imagedeblurring with anti-reflective boundary conditions and re-blurring, Inverse Problems, 22, pp. 2035–2053, 2006.
5. M. Donatelli, S. Serra Capizzano, Anti-reflective boundary conditions and re-blurring,Inverse Problems, 21, pp. 169–182, 2005.
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6. H. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer AcademicPublishers, Dordrecht, The Netherlands, 2000.
7. C.W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Integral Equations ofthe First Kind, Pitman, Boston, 1984.
8. P.C. Hansen, Rank-deficient and discrete ill-posed problems, SIAM, Philadelphia, PA, 1997.9. P.C. Hansen, J. Nagy, D.P. O’Leary, Deblurring Images Matrices, Spectra and Filtering,
SIAM Publications, Philadelphia, 2006.10. M.K. Ng, R.H. Chan, W.C. Tang, A fast algorithm for deblurring models with Neumann
boundary conditions, SIAM J. Sci. Comput., 21, no. 3, pp. 851–866, 1999.11. S. Serra Capizzano, A note on anti-reflective boundary conditions and fast deblurring
models, SIAM J. Sci. Comput., 25–3, pp. 1307–1325, 2003.
Monotonicity of operator with respect to preordering generated by coupling ofsubadditive measures
L. TruffetEcole des Mines de Nantes, Dpt Automatique et Productique, 4, rue Alfred Kastler BP. 20722,
44307 Nantes, Cedex 3, [email protected]
We are studying coupling techniques of particular subadditive measures. These particular mea-sures we are studying are set functions with values in an idempotent semiring (a semiring whichaddition is idempotent). Such measures are used for modelling nonprobabilistic approaches of un-certainity not as a contrast but as a generalization of the probabilistic approach. They are calledpseudomeasures.
Inspired by the probabilistic approach, we define the (increasing) coupling pseudo-measure oftwo pseudo-measures on a dicrete space. Then we derive the necessary and sufficient condition ofthe existence of a coupling measure between two given pseudo-measures. We also chracterize theexistence of an increasing pseudo-measure of two given pseudo-measures as a preorder between thetwo given pseudo-measures.
Next we define and characterize the monotonicity of a linear operator over an idempotent semiringwith respect to the preordering generated by an increasing coupling of pseudo-measures.
Stable rational functions and Hurwitz–like formulas for Hurwitz matrices
Yu. S. Barkovskii∗, M. Yu. Tyaglov†, M. V. Shubnyi∗
∗†Southern Federal University, Rostov-on-Don†Technische Universitat Berlin, Institut fur Mathematik
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Let’s consider rational function
R(z) =h(z)
g(z)= t0z
r−m + t1zr−m−1 + t2z
r−m−2 + . . . . (1)
Here h(z) and g(z) are real polynomials:
h(z) = b0zr + b1z
r−1 + . . . + br−1z + br, b0, b1, . . . , br ∈ R, b0 > 0;
g(z) = c0zm + c1z
m−1 + . . .+ cm−1z + cm, c0, c1, . . . , cm ∈ R, c0 > 0.(2)
and r,m ∈ N⋃0 (r +m > 0).
Definition. Real rational function is called stable if all its roots lye in an open left half-plane ofcomplex plane and all its poles lye in an open right half-plane.
It is proved that
Theorem 1. Real rational function R(z) (1)–(2) is stable iff all successive principal minors ∆k(R)of infinite Hurwitz matrix
H(R) =
t1 t3 t5 t7 t9 . . .t0 t2 t4 t6 t8 . . .0 t1 t3 t5 t7 . . .0 t0 t2 t4 t6 . . .0 0 t1 t3 t5 . . .0 0 t0 t2 t4 . . .. . . . . . . . . . . . . . . . . . . . . .
are positive up to order n = r +m:
∆1(R) > 0, ∆2(R) > 0, . . . , ∆n(R) > 0.
There is the following
Theorem 2. Real rational function R(z) (1)–(2) is stable iff
∇2 > 0, ∇4 > 0, . . . , ∇2n > 0,
where ∇2j are determinants of order 2j formed of coefficients of polynomials h(z) and g(z):
∇2j ≡
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
c0 c1 c2 c3 c4 c5 . . . cj−1 cj . . . c2j−2 c2j−1
b0 b1 b2 b3 b4 b5 . . . bj−1 bj . . . b2j−2 b2j−1
0 0 c0 c1 c2 c3 . . . cj−3 cj−2 . . . c2j−4 c2j−3
0 b0 b1 b2 b3 b4 . . . bj−2 bj−1 . . . b2j−3 b2j−2
0 0 0 0 c0 c1 . . . cj−5 cj−4 . . . c2j−6 c2j−5
0 0 b0 b1 b2 b3 . . . bj−3 bj−2 . . . b2j−4 b2j−3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 0 0 0 0 0 . . . 0 0 . . . c0 c10 0 0 0 0 0 . . . b0 b1 . . . bj−1 bj
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
We assume that bk = 0 whenever k > r and cj = 0 whenever j > m.
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On the Inner and Outer Geometry of the Field of Valuesusing Geometric Computing Methods
Frank UhligDepartment of Mathematics, Auburn University
Auburn, AL 36849–5310, [email protected]
The field of values F (A) = x∗Ax ∈ C | x ∈ Cn, ‖x‖ = 1 of a matrix A ∈ Cn,n is a compact andconvex subset of the complex plane.
We develop an algorithm to decide whether a given point p ∈ C lies inside or outside the fieldof values of a given matrix and that computes the distance, or generalized Crawford number, of anarbitrary point p ∈ C from the boundary of F (A). Moreover for a point p ∈ F (A) we describe afurther algorithm for the inverse problem of finding a unit vector x ∈ Cn with x∗Ax = p.
These algorithms use simple and efficient methods of geometric computing and lead to questionsabout the covering number of points p inside F (A). Some of the questions are answered and othersare formulated as conjectures on the inner geometry of the quadratic map : x ∈ Cn, ‖x‖ = 1 →x∗Ax ∈ C.
SSA-based approach to decomposition of two-dimensional scalar fields
Konstantin Usevich∗, Nina Golyandina†
St.Petersburg State University, Mathematical [email protected]∗, [email protected]†
Singular Spectrum Analysis (SSA) has been approved in many areas as an effective model-free technique of analysis of one-dimensional time series [1,3] (the word “time” is not essentialhere as the method deals with arbitrary equidistant measurements). The SSA approach (an-other name is “Caterpillar” [2]) can be applied to such problems as decomposition of time seriesinto sum of trend, oscillations, and noise, detection of periodicities, smoothing, signal denoising,forecasting extracted time series components, imputation of missing data, change-point detection(see http://www.gistatgroup.com/cat/ (in English) and http://www.gistatgroup.com/gus/ (in Rus-sian)).
In addition, the SSA technique has much in common with a family of signal processing meth-ods that are devoted to signal denoising for damped/undamped sinusoids and estimation of signalparameters.
Let us shortly describe the basic algorithm of SSA applied to the time series FN = (f0, . . . , fN−1).For a given window length L, 1 < L < N , there exists a one-to-one correspondence between sequencesof length N and Hankel L×K-matrices (called trajectory matrices), where K = N −L+1: FN ←→X = (xij)
L,Ki,j=1, xij = fi+j−2. First, we perform the singular value decomposition of the trajectory
matrix constructed from the time series FN . Then we group the SVD matrix components, sum themand obtain a decomposition of the matrix X =
∑i
Xi. Finally, by means of projection on the set
of Hankel (trajectory) matrices in Frobenius norm, we come to a decomposition of the initial timeseries.
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If we solve the problem of signal denoising for finite-rank signals (i.e., signals with rank-deficienttrajectory matrices) by SSA, we should group the d leading SVD components, where d is equal tothe signal rank. The corresponding problem of signal processing is solved via approximation of thetrajectory full-rank Hankel matrix by a low-rank Hankel matrix.
SSA has extensions to simultaneous analysis of several time series (MSSA [2,1]) and to analysisof 2D scalar fields (2D-SSA, [2]).
The 2D-SSA algorithm coincides with the basic (one-dimensional) SSA algorithm up to construc-tion of an analogue of the trajectory matrix. For a given field
F =
f1,1 f1,2 . . . f1,Nc
f2,1 f2,2 . . . f2,Nc
......
. . ....
fNr,1 fNr,2 . . . fNr,Nc
and fixed window sizes (Lr, Lc), 1 ≤ Lr ≤ Nr, 1 ≤ Lc ≤ Nc, 1 < LrLc < NrNc, we introduce twoequivalent representations of the trajectory matrix analogue.
The first representation is the block Hankel matrix with rectangular blocks; let us call it theblock trajectory matrix:
W =
A1 A2 . . . AKr
A2 A3 . . . AKr+1...
.... . .
...ALr ALr+1 . . . ANr
, (1)
where Kr = Nr − Lr + 1 and Ai is the Lc-trajectory matrix for the i-th row of the matrix F:
Ai =
fi,1 fi,2 . . . fi,Kc
fi,2 fi,3 . . . fi,Kc+1...
.... . .
...fi,Lc fi,Lc+1 . . . fi,Nc
,
where Kc = Nc − Lc + 1.Note that the column vectors of the block trajectory matrix (1) can be expressed as the vectorized
moving 2D-windows
Fi,j =
fi,j fi,j+1 . . . fi,j+Lc−1
fi+1,j fi+1,j+1 . . . fi+1,j+Lc−1...
.... . .
...fi+Lr−1,j fi+Lr−1,j+1 . . . fi+Lr−1,j+Lc−1
. (2)
Therefore, the other natural form of W is the so called 2D-trajectory matrix
X =
F1,1 F1,2 . . . F1,Kc
F2,1 F2,2 . . . F2,Kc
......
. . ....
FKr,1 FKr ,2 . . . FKr ,Kc
. (3)
In the above notation, the decomposition step of 2D-SSA (an analogue of the SVD step in theone-dimensional case) consists in the SVD of the block trajectory matrix (1) or, that is equivalent,the tensor Kronecker-Product SVD of the 2D-trajectory matrix (3).
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There are some methods in a sense similar to 2D-SSA. For example, there are results on extractionand parameter estimation for sums of products of damped sinusoids from noisy 2D fields by meansof approximation of a matrix like (1), with no references to SSA [4]. Also, there is a method ofimage processing that uses the KP-SVD of matrices having the form (3); however, the submatricescorresponding to (2) are non-overlapping [5], unlike 2D-SSA.
The aim of this paper is to describe the 2D-SSA approach to 2D fields (images) decomposition,including the transfer of the main SSA concepts (separability, time series of finite rank, continuation)[3] to the two-dimensional case and an overview of related two-dimensional problems that can besolved by 2D-SSA (e.g. smoothing, image/field denoising, image reconstruction, periodic noiseremoval, filling in missing data).
References1. Elsner, J., Tsonis, A., Singular Spectrum Analysis. A New Tool in Time Series Analysis,
New York, Plenum Press, 1996.2. Danilov, D. and Zhigljavsky, A. (Eds.), Principal Components of Time Series: the
“Caterpillar” method, St.-Petersburg, SPbSU Press, 1997. (in Russian).3. Golyandina, N., Nekrutkin, V., and Zhigljavsky, A., Analysis of Time Series Struc-
ture: SSA and Related Techniques, London, Chapman & Hall/CRC, 2001.4. Rouquette, S. and Najim, M., Estimation of frequencies and damping factors by two-
dimensional ESPRIT-type methods, IEEE Transactions on Signal Processing, 49(1) (2001), 237–245.5. Zhang, H. et al., Representing Images by Multiple Kronecker Product Sum. CAD/Graphics,
2003.
Factor rank of non-negative matrices with fixed classical rank
Olga A. VaysmanLomonosov Moscow State University
The theory of non-negative matrices is a classical branch of algebra which is actively developingnowadays, see for example [1] and its references. As regards the matrices over a field, rank is one ofthe most effective invariants. Semiring matrix theory has the same situation with one distinction:there are several different rank functions which can be investigated together or separately. Also ifthe semiring under consideration is embedded in a field we can investigate the difference betweenclassical rank over the field and ranks over the semiring.
Let us consider non-negative real numbers R+. One of the most important in different applica-tions rank functions for matrices over R+ is defined in the following way:
Definition 1. Matrix A ∈ Mm,n(R+), A 6= 0, has factor rank equal to k (f(A) = k) if k is theleast non-negative integer such that there exists a decomposition A = BC where B ∈ Mm,k(R+),C ∈Mk,n(R+). By definition, it is said that f(0) = 0.
It is easy to see that for any matrix A ∈ Mm,n(R+) it holds that rk(A) ≤ f(A). The followingtheorem shows that non-negative matrices with fixed classical rank can have an arbitrary large factorrank. That is, in a certain way there is no dependence between classical and factor rank of matriceswith non-negative real entries.
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Theorem 1. For all natural numbers k ≥ r ≥ 3 there exists a natural number n and a matrixA ∈Mn,n(R+) such that rk(A) = r and f(A) > k where rk(A) denotes the classical rank of A.
I wish to thank my scientific advisor Dr. Alexander E. Guterman for problem statement andhelpful hints.
References1. R. Brualdi and H. Ryser, Combinatorial Matrix Theory, Cambridge University Press,
Cambridge, 1991.
Product/quotient SVD using rank structured matrices
Yvette Vanberghen, Marc Van Barel, Paul Van DoorenKatholieke Universiteit Leuven, Department of Computer Science, Belgium
In this talk we reduce the computation of the singular values of a general product/quotient ofmatrices to the computation of the singular values of an upper triangular semiseparable matrix.Compared to the reduction into a bidiagonal matrix the reduction into semiseparable form exhibitsa nested subspace iteration. Hence, when there are large gaps between the singular values, thesegaps manifest themselves already during the reduction algorithm in contrast to the bidiagonal case.
Band plus Algebra Preconditioners for BTTB Toeplitz Systems
D.Noutsos∗ and P. Vassalos†∗Department of Mathematics, University of Ioannina, GR-451 10 Ioannina, Greece
†Department of Informatics, Athens University of Business and Economics, GR-104 34, Athens,Greece
∗[email protected], †[email protected]
We are concerned with the fast and efficient solution of nm × nm symmetric ill-conditionedblock Toeplitz with Toeplitz blocks (BTTB) linear systems of the form Tnm(f)x = b where thegenerating function f is a priori known and in particular is real valued, nonnegative, having isolatedroots of even order. The preconditioner that we propose is a product of a band Toeplitz matrixand matrices that belong to a certain trigonometric algebra. The underline idea of the proposedscheme, is to embody the well known advantages that each component of the product presents whenit is used alone. As a result we obtain a flexible preconditioner which can be applied to the systemTnm(f)x = b infusing fast convergence to the PCG method. For that, we compare the proposedmethod with techniques already employed in the literature. Our results fully confirm the effectivenessof the proposed strategy and the adherence to the theoretical analysis.
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Iterative technologies for stiff problems with jumping anisotropic diffusion tensors
Yu. Vassilevski∗, S. Goreinov and V. ChugunovInstitute of Numerical Mathematics, Russian Academy of Sciences, 8, Gubkina str., 119991,
Moscow, Russia∗[email protected]
Finite element discretizations of the diffusion equation produce large and stiff systems with sparsematrices. Spectral properties of the matrices deteriorate with the mesh refinement. The degradationis even stronger in the case of jumping and anisotropic diffusion tensors. Several blackbox techniquesfor the iterative solution of the above systems are reviewed. Numerical comparisons for a 3D modelproblem are presented.
An Enestrom-Kakeya theorem for hermitian polynomial matrices
Harald K. WimmerMathematisches Institut, Universitat Wurzburg, D-97074 Wurzburg, Germany
The Enestrom-Kakeya theorem states that a real polynomial h(z) = amzm + · · ·+ a1z+ a0 with
am ≥ · · · ≥ a1 ≥ a0 ≥ 0, am > 0, has all its zeros in the closed unit disc and that the zeros on theunit circle are simple. We extend the Enestrom-Kakeya theorem and its refinement by Hurwitz topolynomial matrices H(z) with positive semidefinite coefficients. We determine an annular regioncontaining the zeros of detH(z). A stability result for systems of linear difference equations is givenas an application. (Joint work with G. Dirr.)
Positivity criteria generalizing the leading principal minors criterion
Nadya ZharkoMech.-Math. Faculty, Kiev National University, Vladimirskaya 64, Kiev, Ukraine
This is joint work with Vyacheslav Futorny and Vladimir V. Sergeichuk.An n×nHermitian matrix is positive definite if and only if all leading principal minors ∆1, . . . ,∆n
are positive. We show that certain sums δl of l× l principal minors can be used instead of ∆l in thiscriterion. We describe all suitable sums δl for 3 × 3 Hermitian matrices. For an n × n Hermitianmatrix A partitioned into blocks Aij with square diagonal blocks, we prove that A is positive definiteif and only if the following numbers σl are positive: σl is the sum of all l × l principal minors thatcontain the leading block submatrix [Aij ]
k−1i,j=1 (if k > 1) and that are contained in [Aij ]
ki,j=1, where
k is the index of the block Akk containing the (l, l) diagonal entry of A. We also prove that σl canbe used instead of ∆l in other inertia problems.
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References1. V. Futorny, V.V. Sergeichuk, N. Zharko, Positivity criteria generalizing the leading
principal minors criterion, Positivity 11 (1) (2007) 191–199.
The Method of Regularization of Tikhonov Based on Augmented Systems
A. I. Zhdanov∗, T.G. ParchaikinaSamara State Aerospace University, Moskovskoe sh. 34, Samara, 443086 Russia
Formulation of the problem. Solving an approximate systems of linear algebraic equations(SLAEs) is fundamental problem of numerical analysis.
Let the exact SLAEAu = f (1)
be specified by the (a priori unknown) initial data d = A, f, where A = (aij) ∈ Rm×n, f =(f1, . . . , fm)⊤ ∈ Rm and u = (u1, . . . , un)⊤ ∈ Rn.
In the general case, the “solution” to exact SLAE (1) is understood as its normal pseudosolution
u∗ = A+f, (2)
where A+ is the pseudoinverse or the Moore–Penrose generalized of A.Then, the inconsistency measure of the exact SLAE (1) is defined by
µ = infu∈Rn
‖Au− f‖ = ‖Au∗ − f‖ ≥ 0.
Throughout the report, the vector norms in Rm and Rn are the Euclidean (quadratic) norms; i.e.,‖r‖ = ‖r‖2, where r = f −Au is the residual.
The information on system (1) is given by the approximate data d = A, f (i.e., by an individualapproximate SLAE Au = f) such that
‖A−A‖ ≤ h, ‖f − f‖ ≤ δ,
where the scalars h ≥ 0 and δ ≥ 0 specify the errors in the assignment of the approximate data dand ‖A‖ is the spectral norm of A.
If rankA < min(m,n), then solving system (1) (as specified in (2)) on the basis of approximatedata d with h > 0 is an ill-posed problem in the sense of Hadamard, because the approximate normalpseudosolution
u∗ = A+f
is unstable for infinitesimal perturbations in the initial data.To determine stable solutions to system (1) from approximate data d, various regularization
methods are used. The regularization method of A.N. Tikhonov is among the most universal ones.It is well known that, in this method, the regularized solution uα is determined as a (unique) solutionto the Euler equation
(A⊤A+ αEn)u = A⊤f , α > 0, (3)
93
where En is the identity matrix of order n and α is the regularization parameter.Method of augmented systems. In this report, a approach is proposed for systems with
µ > 0. This approach is based on the reduction of the original system to an equivalent consistentaugmented system. Moreover, the proposed approach makes it possible to obtain efficient numericalalgorithms for solving the problem under discussion.
The normal system of equations is equivalent to the augmented system
r + Au = f ,
A⊤r = 0⇐⇒
(Em A
A⊤ 0
)(ru
)=
(f0
)⇐⇒ Gz = b. (4)
where z = (r⊤, u⊤)⊤ ∈ Rm+n.Hence, the regularized solution zα to system (4) is determined as a (unique) solution to the Euler
equation(G2 + αEm+n)z = Gb. (5)
Thus, using equivalent augmented systems, we can directly extend the basic results obtained forconsistent systems (µ = 0) to the class of inconsistent systems (µ > 0).
Using the symmetry of G, we can reduce the condition number of the regularized (5). To thisend, we apply the method of an imaginary shift of the spectrum (Faddeeva’s method).
According to this method, we replace Eq. (5) by the equation
(G+ i√α · E)z = b, (6)
where i =√−1 is the imaginary unit. An immediate implication of (6) is that x = Re z is a solution
to the equation (5).Condition number of system (6) is estimated by inequality
cond2(G+ i√αE) ≤
√
1 +1
4α
(1 +
√1 + 4σ2
1
)2
,
where σ1 is the maximal singular value of A.
References1. Zhdanov A. I., Regularization of Unstable Finite-Dimensional Linear Problems Based on
Augmented Systems, Comp. Math. and Math. Physics, Vol. 45, No. 11, 2005, pp. 1845–1853.
The radioactive pollution transport modeling in airnearby Volgodonsk nuclear station
V. N. Zubov,∗ L. A. Krukier†, G. V. Muratova, T. N. Subbotina‡
Computer Center of Southern Federal University, Rostov-on-Don, Russia∗[email protected], †[email protected], ‡[email protected]
The importance of the atmosphere radio nuclear pollution modeling directly connected with thedevelopment of nuclear industry. Like in many other regions of Russia, the lack of hydrocarbonenergy exists in the North Caucasus. Hence Volgodonsk nuclear station (VNS) has great economical
94
value in the South of Russia. The first module of APP was included into the common power grid in2001, and in June of 2006 the decision of building the second module was taken.
At the same time VNS brings extra risks in the ecosystem of the region. And such risks are asimportant as ordinary energy supply problems. Thus radioactive safety around the nuclear stationand its environmental territory is of great importance.
Modeling of different out of order and extreme situation in atmosphere nearby nuclear stationcan be considered as investigation of large-scale transport processes with point source of studiedsubstance. This problem can be divided into several parts: prediction of pollution movement in air,deposition on water and soil, analysis of possible effects on activity of population. The predictionsupposes the possibility of taking decisions immediately with certain accuracy.
Each model of pollution transport consists of two main components:Meteorological model of the wind fieldTransport model which determines concentration of considered substance according to diffusion,
advection, deposition, interaction between substances and other important processes.The main purpose in modeling of radioactive pollution transport is estimation of radioactive
influence risks. Thus it is rather convenient to use activity of radioactive substances instead of itsquantity in calculations.
Model of radionuclide transport in atmosphere is based on 3-dimensional turbulent diffusionequation in Cartesian coordinates. The medium can be considered as incompressible with regard topermanency of air density in the domain of calculations.
∂ϕ∂t = −1
2
(v1
∂ϕ∂x + ∂(v1ϕ)
∂x + v2∂ϕ∂y + ∂(v2ϕ)
∂y + v3∂ϕ∂z + ∂(v3ϕ)
∂z
)+
+Kx∂2ϕ∂x2 +Ky
∂2ϕ∂y2 + ∂
∂z
(Kz
∂ϕ∂z
)+
+E (x, y, z, t)− vdϕ− Λϕ− krϕ,
ϕ|t=0 = ϕ0 (x, y, z) , β ϕ+ γ ∂ϕ∂n
∣∣∣∂D
= ϕΓ (s, t) , s ∈ ∂D.
The results received in numerical experiments allow to estimate ecological situation nearby VNSworking in regular and irregular regimes.
Experiments can be applied in taking precautions against unfavorable influence of radio nuclearsubstances and making a shield plan in case of unpredictable situations.
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