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Insights into block rational Krylov methods
Stefan Guttel(with Steven Elsworth and Kathryn Lund)
The University of Manchester
Stefan Guttel Insights into block rational Krylov methods 1 / 16
Block Krylov methods: not a new idea
In: Proceedings of the IEEE Conference on Decision and Control, pp. 505–509, 1974.
Stefan Guttel Insights into block rational Krylov methods 2 / 16
Motivation: why do we need block Krylov methods?
Let A ∈ CN×N be Hermitian and b ∈ CN nonzero. Let p be thesmallest-degree monic polynomial such that
p(A)b = 0.
Then in exact arithmetic the non-block Lanczos algorithm will locate the(simple) roots λ of p and find one eigenvector v such that Av = λv.
=⇒ multiple eigenvalues or tight clusters will not be found (quickly).
Stefan Guttel Insights into block rational Krylov methods 3 / 16
Convergence of non-block Ritz values is slow and erraticExample: A = wilkinson(100) and b = randn(100,1).Plot the Ritz values of orders j = 1, 2, . . . , 100.Colour indicates the distance to a closest eigenvalue of A.
order j
eige
nval
ue in
dex
s = 1
10 20 30 40 50 60 70 80 90
10
20
30
40
50
60
70
80
90
100 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stefan Guttel Insights into block rational Krylov methods 4 / 16
Convergence of non-block Ritz values is slow and erraticExample: A = wilkinson(100) and b = randn(100,1).Plot the Ritz values of orders j = 1, 2, . . . , 100.Colour indicates the distance to a closest eigenvalue of A.
order j
eige
nval
ue in
dex
s = 1
10 20 30 40 50 60 70 80 90
10
20
30
40
50
60
70
80
90
100 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stefan Guttel Insights into block rational Krylov methods 4 / 16
Convergence of block Ritz values is more reliableExample: A = wilkinson(100) and b = randn(100,2).Plot the block Ritz values of orders j = 1, 2, . . . , 50.Colour indicates the distance to a closest eigenvalue of A.
order j
eige
nval
ue in
dex
s = 2
5 10 15 20 25 30 35 40 45
10
20
30
40
50
60
70
80
90
100 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stefan Guttel Insights into block rational Krylov methods 5 / 16
Block methods can speed up computationsMulti-source forward simulation of a synthetic 3D hydrocarbon reservoirrequires repeated integrations of large-sparse linear ODEs
Me′(t) = Ke(t), e(0) = ei ∈ RN
with multiple initial conditions e1, . . . , es . Here, N = 810 100 and s = 45.
Stefan Guttel Insights into block rational Krylov methods 6 / 16
Block methods can speed up computations ctd.
Define b := [e1, . . . , es ] and compute M-orthonormal basis Vm ∈ CN×ms ,
VTmMVm = Ims , of a block rational Krylov space
Vm :=
m∑j=1
(K − ξjM)−1bCj : Cj ∈ Cs×s
⊂ RN×s
using (distinct) optimized shifts ξ1, . . . , ξm > 0.
Rational Arnoldi approximation
e(t) := Vm exp(t VTmKVm)(VT
mei )
approximately solves Me′(t) = Ke(t), e(0) = ei .
Stefan Guttel Insights into block rational Krylov methods 7 / 16
We measured 1.5x speedup with block vs non-block
[Qiu/Guttel/Ren/Yin/Liu/Zhang/Egbert, Geophys. J. Int., 2019]
Stefan Guttel Insights into block rational Krylov methods 8 / 16
Speedup depends on application and architecture
Some block speedup factors reported in literature:
Block FGMRES-DR solver for adjoint problems in aerodynamic designoptimization in [Pinel/Montagnac, AIAA J., 2013]: 1.1–1.4
Block QMR/GMRES solvers for EM modelling in frequency domainin [Puzyrev/Cela, Geophys. J. Int., 2015]: 1.5–2.0
Block GMRES with recycling in [Parks/Soodhalter/Szyld, 2016]:about 1.5 (matvec products only)
Block FOM solver for variational data assimilation in [Mercier et al.,Q. J. R. Meteorol. Soc., 2018]: about 5 (per iteration)
Block CG solver on GPUs for lattice QCD in [Clark et al., Comp.Phys. Comm., 2018]: about 2–5 (using mixed precision)
=⇒ Block speedups of up to 5x appear to be realistic.
Stefan Guttel Insights into block rational Krylov methods 9 / 16
But it’s not always about speed!
There might be “structural” requirements to use block methods:
Eigenvalue problems with clustered eigenvalues:Mehrmann/Schroder 2011, Freitag/Kurschner/Pestana 2018,Drineas/Ipsen/Kontopoulou/Magdon-Ismails 2018
Matrix equations with low-rank right-hand sides:El Guennouni/Jbilou/Riquet 2002, Heyouni/Jbilou 2009
Model order reduction of MIMO systems:Freund 2000, Abidi/Hached/Jbilou 2014
Block Krylov methods have proven to be useful in practice.
Our recent focus (jointly with Steven Elsworth):
analyze premature breakdowns of block rational Arnoldi algorithm;
provide robust implementations of block rational Krylov methods.
Stefan Guttel Insights into block rational Krylov methods 10 / 16
But it’s not always about speed!
There might be “structural” requirements to use block methods:
Eigenvalue problems with clustered eigenvalues:Mehrmann/Schroder 2011, Freitag/Kurschner/Pestana 2018,Drineas/Ipsen/Kontopoulou/Magdon-Ismails 2018
Matrix equations with low-rank right-hand sides:El Guennouni/Jbilou/Riquet 2002, Heyouni/Jbilou 2009
Model order reduction of MIMO systems:Freund 2000, Abidi/Hached/Jbilou 2014
Block Krylov methods have proven to be useful in practice.
Our recent focus (jointly with Steven Elsworth):
analyze premature breakdowns of block rational Arnoldi algorithm;
provide robust implementations of block rational Krylov methods.
Stefan Guttel Insights into block rational Krylov methods 10 / 16
The block rational Arnoldi algorithm
Input: A ∈ CN×N , b ∈ CN×s , shifts ξ1, . . . , ξm ∈ C \ Λ(A).
1. v1 := bR−1 such that v∗1v1 = Is2. for j = 1, . . . ,m do3. Choose a continuation block vector tj ∈ Cjs×s
4. w := (A− ξj I )−1Vjtj
5. for i = 1, . . . , j do6. Ci ,j := v∗i w7. w := w − viCi ,j
8. end for9. vj+1 := wC−1
j+1,j such that v∗j+1vj+1 = Is10. Vj+1 := [Vj , vj+1]11. end for
Continuation block vector tj affects the numerical rank of [Vj ,w]!
Stefan Guttel Insights into block rational Krylov methods 11 / 16
Block rational Arnoldi decompositions
The rational Arnoldi algorithm produces decompositions of the form
AVm+1Km = Vm+1Hm
with block upper-Hessenberg matrices Km and Hm:
A v1 v2 v3
K11K12
K21K22
K32 v1 v2 v3
H11H12
H21H22
H32=
Stefan Guttel Insights into block rational Krylov methods 12 / 16
An integral formula for the block basis vectors
Associated with each basis block vector vj+1 is a rational function
Rj : C→ Cs×s such that
vj+1 = Rj(A) ◦ v1 :=1
2πi
∫Γ(zI − A)−1v1Rj(z)−1dz ,
with a contour Γ including Λ(A) and excluding the poles of Rj .
[Elsworth & G., MIMS Eprint 2019.2, 2019]
Stefan Guttel Insights into block rational Krylov methods 13 / 16
A connection with nonlinear eigenvalue problems
Theorem (Elsworth & G., 2019)
The points λ ∈ C where Rj(λ) is singular are contained in Λ(Hj ,Kj).
In order to avoid premature breakdown of the rational Arnoldi algorithmin the step w := (A− ξj I )
−1Vjtj , the continuation block vector tj should
be chosen such that the rational block function R(z) satisfying
Vjtj = R(A) ◦ v1
is nonsingular at ξj .
Can show: A block left null vector t∗j such that t∗j (Hj−1 − ξjKj−1) = 0∗
satisfies conditions of the theorem, generalizing a result by Ruhe (1998).
=⇒ our new recommendation for the continuation block vector tj .
Stefan Guttel Insights into block rational Krylov methods 14 / 16
INLET test from Oberwolfach CollectionProblem: Compute rational Krylov basis for nonsymmetric A,E ∈ RN×N
and a block starting vector b ∈ RN×2, where N = 11 730.
xi1 = 1i*repmat(linspace(0, 40, 4), 1, 6);
xi2 = xi1;
xi2(13) = 0.996 - 0.0762i;
The 13th pole is changed to 0.996000− 0.0762000i , which is close toan eigenvalue 0.996026− 0.0762341i of the matrix pencil (H12,K12).
Results:
xi1 xi2
'last' 'ruhe' 'last' 'ruhe'
cond 8.7× 105 4.2× 103 1.4× 109 4.8× 104
space 5.0× 10−6 1.5× 10−11 7.3× 10−3 1.9× 10−10
Stefan Guttel Insights into block rational Krylov methods 15 / 16
Summary
Block (rational) Krylov applications for eigenvalue computations,block linear systems, matrix equations, in model order reduction, etc.
Block speedups of up to 5x appear to be realistic in practice.
Use of block methods might be imposed by problem structure.
Connection rational Arnoldi method ⇐⇒ nonlinear eigenproblems.
New continuation strategy as default in Rational Krylov Toolbox,available at
http://rktoolbox.org
———S. Elsworth and S. Guttel. The block rational Arnoldi method.Submitted 2019. http://eprints.maths.manchester.ac.uk/2685/
Stefan Guttel Insights into block rational Krylov methods 16 / 16