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Using the Tensor-Train approach to solve the
ground-state eigenproblem for hydrogen molecules
Alexander Veit∗ L. Ridgway Scott†
Abstract
We consider the Born-Oppenheimer approximation of the Schrödinger equationfor two hydrogen atoms in the case of large separation distances. We show thatthe Feschbach-Schur perturbation method can be used to solve the problem for thedierence between a given separation and innite separation. This leads to a sim-plied problem which can be solved iteratively. We show that this iteration con-verges for suciently large separation distances and we solve the arising sequence ofsix-dimensional PDEs with a Finite Element Method in combination with low-ranktensor techniques to make the computations tractable. In particular we show howthe discretized problems can be represented and solved in the Tensor-Train format.Since the storage and computational complexity of this format scale linearly in thedimension, a very large number of grid points can be employed which leads to ac-curate approximations of the ground-state energy and ground-state wavefunction.Various numerical experiments show the performance and accuracy of this method.
AMS subject classications: 65N30, 15A69
Keywords: Schrödinger equation, Born-Oppenheimer approximation, low-ranktensor approximation, Tensor-Train format
1 Introduction
The interaction energy between two hydrogen atoms can not determined in closed form.The simplest quantum-mechanical model for this system is based on the Schrödingerequation in six-dimensional space, using the Born-Oppenheimer approximation to elim-inate degrees of freedom associated with the hydrogen nuclei. Conventional methodsusing nite dierence or nite element methods to solve the eigenvalue problem for theSchrödinger partial dierential operator are prohibitive due to the high dimension.It is well-known that the ground state of the hydrogen molecule is a singlet spin state [19],and this allows a reduction to solving an eigenvalue problem for the Schrödinger operatorwithout any symmetry or anti-symmetry considerations for the spatial variables [7]. Inother problems, there would be a need to include certain algebraic conditions related tothe Pauli exclusion principle.
∗Department of Computer Science, University of Chicago, Chicago, Illinois 60637, USA. Supportedby SNSF grant P2ZHP2_148705.†Departments of Computer Science and Mathematics, The Computation Institute, and the Institute
for Biophysical Dynamics, University of Chicago, Chicago, Illinois 60637, USA. Partially supported byNSF grant DMS-1226019.
1
The state-of-the-art methods for solving the Schrödinger equation use Galerkin methodswith Gaussian functions as the basis for the approximating spaces. Gaussians have specialalgebraic properties that make the resulting approximation algorithmically more ecient.However, the eigenfunctions of the Schrödinger operator decay like e−|x| and not e−|x|
2,
so the Gaussian approximation performs best when atoms are close.When atoms are far apart, we can imagine using a perturbation argument [7] to solvefor the dierence between a given separation and innite separation. We will explorethe Feshbach-Schur perturbation method [15] as an algorithm to solve such problems in aHilbert-space setting. This approach simplies the problem substantially for large separa-tion distances, but it does not reduce the dimensionality of the problem. Thus we also uselow-rank tensor methods to make the computations tractable. In particular we will solvethe simplied problems using a Finite Element Method with piecewise linear polynomialsby representing the Galerkin matrix and the right-hand side in a low-parametric tensor for-mat and by solving the arising linear system within this format. In this way also very nesix-dimensional grids and therefore a very large number of degrees of freedom are feasibleand can be used to solve the simplied problems accurately. Numerical methods based onlow-rank tensor representations gained a lot of attention in recent years and were appliedto a variety of problems including electronic structure calculations, the Fokker-Planckequation, stochastic and parametric PDEs and high-dimensional Schrödinger equations.We refer to [14, 25, 17] for an overview of the available literature on tensor methods.In this paper we use the Tensor-Train format (TT-format) introduced in [28, 27] to repre-sent the Galerkin matrix and the right-hand side of our problem. This format utilizes anSVD-based compression of a given tensor and the required memory to store the approx-imation scales linearly in the dimension. Furthermore operations in the TT-format (e.g.matrix-vector multiplication, addition) can also be performed with linear dependence onthe dimension of the tensor. Advanced algorithms like cross approximation schemes ([29])and solver for linear systems in the TT-format ([11, 12]) are available and implementedin the TT-Toolbox (MATLAB) by I. Oseledets that will be used for the numerical exper-iments in Section 5. There we test the performance and accuracy of our algorithm andcompute the ground-state energy and ground-state wavefunction for dierent separationsdistances of the hydrogen nuclei.We will use atomic units so that ~ = 1, e = 1, me = 1 and 4πε0 = 1, where ~ is thereduced Planck constant, me the mass of the electron, e the elementary charge, and ε0the dielectric permittivity of the vacuum. In this system of units, the length unit is theBohr (about 0.529 Ångstroms) and the energy unit is the Hartree (about 4.36 × 10−18
Joules).
2 Interaction between two hydrogen atoms
2.1 Problem statement
In the following we are interested in the ecient numerical solution of the Born-Oppenheimerapproximation of the Schrödinger equation for a pair of hydrogen atoms separated by adistance 2R. We assume that the nucleus of the rst hydrogen atom is positioned at−Re and the nucleus of the second atom is xed at Re, where e (‖e‖ = 1) is the unitvector pointing from one hydrogen nucleus to the other. Using atomic units throughout
2
the paper the eigenvalue problem can be written as
HRΨR = λRΨR, (2.1)
where HR is the Hamiltonian operator, ΨR is the ground-state wavefunction, satisfyingthe normalization constraint ∫
R3×R3
|ΨR(x, y)|2 dxdy = 1,
and the eigenvalue λR is the ground-state energy. The Hamiltonian in our case can bewritten as
HR = H∞ +WR, (2.2)
whereH∞w = −1
2∆xw −
1
2∆yw −
w
|x+Re|− w
|y −Re|and the correlation potential WR satises
WR(x, y) = − 1
|x−Re|− 1
|y +Re|+
1
|x− y|+
1
2R.
The potentialWR contains all the terms that model the interaction between the hydrogenatoms. Therefore the reduced eigenproblem
H∞ψR = λ∞ψR (2.3)
can be interpreted as a decoupled problem without interactions between the two atoms,as if they were separated by an innitely large distance. Problem (2.3) or more preciselyits solution will be of major importance in the following. It can be shown ([31]) that itadmits an analytic solution which is given by the eigenpair
ψR(x, y) =1
πe−|x+Re|−|y−Re|, λ∞ = −1.
It can be shown (see [7]) that for large R the solution of our original problem (2.1)is a perturbation of the solution of (2.3) in the sense that ΨR = αRψR + yR, where‖yR‖L2(R6) = O
(R−3
)and αR ∈ R. We will discuss this in more detail in Section 2.2.
We now make a change of variable that will simplify the numerical approximation andthe implementation. Let
x 7→ x+Re, y 7→ y −Re. (2.4)
Using the coordinates (2.4) the original eigenvalue problem (2.1) transforms to
HRΨR = λRΨR, (2.5)
where ΨR(x, y) = ΨR(x−Re, y−Re) for (x, y) ∈ R6 and the same normalization condi-tion, i.e. ‖ΨR‖L2(R6) = 1, holds. The modied operator HR can be expressed as
HR = H0 + IR,
3
whereH0w = −1
2∆xw −
1
2∆yw −
w
|x|− w
|y|and
IR(x, y) = − 1
|x− 2Re|− 1
|y + 2Re|+
1
|x− y − 2Re|+
1
2R.
Note that the modied reduced eigenproblem
H0ψ0 = λ0ψ0 (2.6)
has the solutionψ0(x, y) =
1
πe−|x|−|y|, λ0 = −1.
The advantage of the shifted problem (2.5) is that the solution ΨR is exponentially de-caying (as is ΨR) and is furthermore concentrated around the origin. A suitable com-putational domain for the numerical approximation scheme will therefore be simply asix-dimensional cube around the origin.
The interaction energy between the two hydrogen atoms is given by the dierence λR−λ0.It can be shown (e.g. [26, 1]) that this quantity can be accurately approximated by
λR − λ0 ≈ −C6 · (2R)−6 − C8 · (2R)−8 − C10 · (2R)−10, (2.7)
where we use C6 = 6.4990267, C8 = 124.39908 and C10 = 1135.21404 as computed in[8]. This analytic result will be useful to validate our numerical scheme and to check theaccuracy of the approximated ground states.
2.2 Application of the Feschbach-Schur method
Our numerical method is based on the Feschbach-Schur perturbation method. We referto [16] for a general introduction and to [7] for a more detailed description how to applyit to the hydrogen molecule.The application of the Feschbach-Schur method to (2.5) is motivated by the fact thatthe solution ΨR can be considered (at least for suciently large R) as a perturbation ofψ0. We begin by expressing ΨR as a linear combination of ψ0 and the function YR whichrepresents the perturbation. Let
ΨR = XR + YR, (2.8)
where XR = αRψ0. Due to the normalization condition ‖ΨR‖L2(R6) = 1 the constant αRcan be easily determined as
αR =√
1− ‖YR‖2L2(R6).
In order to derive equations for YR and λR which will lead with (2.8) to a solution of (2.5)we dene P0 to be the projection onto the one-dimensional subspace spanned by ψ0,
P0 := |ψ0 >< ψ0|, P0f(x, y) =
∫R6
ψ0(x′, y′)f(x′, y′)dx′dy′ · ψ0(x, y)
4
and furthermore denote P⊥0 := 1 − P0. We now represent ΨR as a vector with twocomponents, i.e.,
Ψ =
(P0Ψ
P⊥0 Ψ
)=
(XR
YR
).
Then the operator HR admits the block form
HR =
(P0HRP0 P0HRP
⊥0
P⊥0 HRP0 P⊥0 HRP⊥0
)
and the eigenvalue problem HRΨR = λRΨR becomes
(P0HRP0 − λR)XR + P0HRP⊥0 YR = 0 (2.9)
P⊥0 HRP0XR + (H⊥R − λR)YR = 0, (2.10)
where H⊥R := P⊥0 HRP⊥0 . Since H0ψ0 = λ0ψ0 we have
P⊥0 HRP0 = |P⊥0 (H0 + IR)ψ0 >< ψ0| = |P⊥0 IRψ0 >< ψ0|,
equation (2.10) leads with the denition of H⊥R and P⊥0 to an equation for YR:(H⊥R − λR
)YR =
(P⊥0 HR − λR
)YR
= −P⊥0 IRψ0
= −IRψ0 + P0IRψ0
= −IRψ0 + εRψ0, (2.11)
where
εR := (ψ0, IRψ0) = e−4R
(1
2R+
5
8− 3R
2− 2R2
3
).
The last formula was established in [32] and recently studied in [5] and veried in [6]. Inorder to obtain an equation for λR we solve the system (2.9)-(2.10) for XR and get
(P0HRP0 − λR)XR − P0HRP⊥0 (H⊥R − λR)−1P⊥0 HRP0XR = 0.
It is shown in [7] that this can be reduced to the scalar equation
λR − λ0 = εR + (IRψ0, YR) . (2.12)
The task of solving the original eigenvalue problem (2.5) is hence reduced to solve (2.11)for the perturbation YR and using formula (2.8) to obtain the ground-state wavefunction.The corresponding eigenvalue can then be computed using (2.12).
In order to nd a suitable computational scheme to solve (2.11) we remark that using thedenition of H⊥R and the fact that P⊥0 YR = YR the equation can be re-written as
(H⊥0 − λR)YR + P⊥0 IRYR + IRψ0 − εRψ0 = 0.
5
Subtracting λ0YR on both sides leads to the problem
(H⊥0 − λ0)YR = (λR − λ0)YR − P⊥0 IRYR − IRψ0 + εRψ0,
λR = λ0 + εR + (IRψ0, YR) .
In order to nd an approximation to this problem we propose the following simple iterativescheme:
(H⊥0 − λ0)Y k+1R = (λkR − λ0)Y k
R − P⊥0 IRY kR − IRψ0 + εRψ0 with Y k+1
R ⊥ ψ0, (2.13)
λk+1R = λ0 + εR +
(IRψ0, Y
k+1R
)(2.14)
for k = 0, 1, . . .. A key feature of this algorithm is that the operator H⊥0 − λ0 doesnot depend on k and hence only a xed operator has to be inverted at each step of theiteration. Furthermore it is important that the correlation potential IR, which is dicultto handle numerically, only appears on the right-hand side of the problem.The iteration (2.13), (2.14) can be initialized by solving
(H⊥0 − λ0)Y 0R = −R−3Bψ0 with Y 0
R ⊥ ψ0 (2.15)
λ0R = λ0 + εR +
(IRψ0, Y
0R
), (2.16)
whereB(x, y) = x · y − 3(x · e)(y · e).
Note that equation (2.15) models the limiting behavior of the perturbation YR as R→∞(see [7]) which makes it a good candidate for the computation of an initial guess of theiteration. Alternatively, if we are interested in several values of R and already havecomputed an approximation of YR, denoted Y K
R then we can initialize the iteration forYR′ where R′ < R with Y K
R .In the next section we prove that a modied version of the above iteration converges forsuciently large R. In Section 2.4 we derive an equivalent problem to (2.13) that can besolved using a Finite Element Method.
2.3 Convergence of the iteration
In order to prove a convergence result for iteration (2.13) we dene a cuto version of IRby
IR := χRIR with χR(x, y) =
1 if ‖(x, y)‖2 ≤ R0 otherwise.
and note that there exists C > 0 such that ‖IR‖L∞(R6) ≤ C/R. Next we dene aperturbed version of (2.13):
(H⊥0 − λ0)Y k+1R = (λkR − λ0)Y k
R − P⊥0 IRY kR − IRψ0 + εRψ0 with Y k+1
R ⊥ ψ0, (2.17)
whereλkR − λ0 = εR + (IRψ0, Y
kR).
6
as in (2.14). Note that both IR and IR are present in the iteration above. The iterationcan be initialized by solving
(H⊥0 − λ0)Y 0R = −R−3Bψ0 with Y k+1
R ⊥ ψ0, (2.18)
λ0R = λ0 + εR + (IRψ0, Y
0R). (2.19)
In the following we will prove that the iteration (2.17) together with (2.18) and (2.19)converges for suciently large R. The following lemma shows that the iterates Y k
R remainbounded if R is large enough.
Lemma 2.1. It holds
‖Y kR‖2 := ‖Y k
R‖L2(R6) ≤ 1, ∀k ∈ N0,
if R satises
C0(C1e−R + 2C2R−3 + C3R
−1 + C1e−R) ≤ 1, (2.20)
with constants
C0 = ‖(H⊥0 − λ0)−1‖ ‖IRψ0‖2 ≤ C2R−3
|εR| ≤ C1e−R ‖IR‖∞ = C3R−1.
Proof. For k = 0 we have‖Y 0
R‖2 ≤ C0R−3‖Bψ0‖2 ≤ 1
for R satisfying (2.20). Assume now that ‖Y kR‖ ≤ 1. The estimates in [7] show that
‖Y k+1R ‖2 ≤ C0
(|λkR − λ∞|‖Y k
R‖2 + ‖P⊥0 IRY kR‖2 + ‖IRψ0‖2 + ‖εRψ0‖2
)≤ C0(C1e−R + C3R
−1 + 2C2R−3 + C1e−R)
≤ 1.
The statement follows by induction.
We furthermore need that the Born-Oppenheimer Hamiltonian has a well localized groundstate. More precisely the following Theorem holds (see [1, Theorem 1.0.2], also [16,Chapter 8,10,12]).
Theorem 2.2. The operator HR has an innite number of eigenvalues, λj, below its
essential spectrum σess(HR). Moreover, the corresponding eigenfunctions Ψj satisfy the
estimate
|Ψj(x, y)| ≤ CRe−α|(x,y)| (x, y) ∈ R6,
where α ≤ Σ− λj and λj < Σ := inf σess(HR).
Theorem 2.2 leads to the following result for the shifted operator HR.
Corollary 2.3. Let R ∈ [Rmin, Rmax] with Rmax < ∞. There exist C,α > 0 such that
the eigenfunctions of the operator HR in the coordinates (2.4) satisfy
|Ψj(x, y)| ≤ Ce−α|(x,y)| (x, y) ∈ R6,
where C is independent of R.
7
With the results above we can establish the following theorem.
Theorem 2.4. Let the assumptions of Lemma 2.1 be satised. Then the iteration (2.17)together with (2.18) and (2.19) converges for suciently large R and the error EkR =YR − Y k
R satises
‖Ek+1R ‖L2(R6) ≤
(C
R
)k+1
‖E0R‖L2(R6) + e−R
R
R− C.
for some C < R.
Proof. In the following C denotes a constant that can change its value from line to line.With EkR = YR − Y k
R we get
(H⊥0 − λ0)Ek+1R = (λR − λkR)YR + (λkR − λ0)EkR − P⊥0 IREkR − P⊥0 (IR − IR)YR
and thus
‖Ek+1R ‖2 ≤ C
(|λR − λkR|‖YR‖2 + |λkR − λ0|‖EkR‖2 + ‖IREkR‖2 + ‖(IR − IR)YR‖2
).
(2.21)We will subsequently bound each term in (2.21). In [7] it is shown that ‖IRψR‖2 ≤ CR−3
and ‖YR‖L2(R6) ≤ CR−3. We therefore get
|λR − λkR| = (IRψR, EkR) ≤ ‖IRψR‖2‖EkR‖2 ≤ CR−3‖EkR‖2
and|λR − λkR|‖YR‖2 ≤ CR−6‖EkR‖2.
For the second term we get with Lemma 2.1 that
|λkR − λ0| ≤ Ce−R + ‖IRψR‖2‖Y kR‖2
≤ CR−3.
For the third term we remark that ‖IR‖L∞(R6) ≤ C/R. Therefore
‖IREkR‖2 ≤C
R‖EkR‖2.
For the last term we have
‖(IR − IR)YR‖L2(R6) = ‖IRYR‖L2(BcR), (2.22)
where BCR = R6\BR and BR is a ball of radius R. In order to estimate (2.22) we note that
Corollary 2.3 shows that the ground-state wave function of our original problem satises|ΨR(x, y)| ≤ Ce−α|(x,y)|. Since ΨR = νRψ0 + YR with νR =
√1− ‖YR‖22, we also have
|YR(x, y)| ≤ Ce−α|(x,y)|.
Similar to [7] we can therefore show that
‖IRYR‖L2(BcR) ≤ CRe−R.
8
Collecting all of the above error terms shows that there exists C > 0 and C > 0 s.t.
‖Ek+1R ‖2 ≤ C
(R−6‖EkR‖2 +R−3‖EkR‖2 +R−1‖EkR‖2 +Re−R
)≤ C
(R−1‖EkR‖2 +Re−R
).
Let R be such that C < R. Then
‖Ek+1R ‖2 ≤
(C
R
)k+1
‖E0R‖2 + e−R
R
R− C.
The iteration thus converges for suciently large R up to an error of e−R RR−C which is
due to the cuto of IR.
2.4 An approximation using FEM
In order to solve equation (2.13) we set Y k+1R = P⊥0 Z
k+1R and solve for Zk+1
R . Note thatthe operator is H0 is self adjoint, i.e.,
(H0v, w)L2(R6) = (v,H0w)L2(R6)
which implies
P0H0w = (H0w,ψR)L2(R6)ψR = (w,H0ψR)L2(R6)ψR = λ0P0w (2.23)
for suciently smooth w. Let
fkR := (λkR − λ0)Y kR − P⊥0 IRY k
R − IRψ0 + εRψ0
denote the right-hand side of (2.13). Then solving (2.13) is equivalent to
(H⊥0 − λ0)P⊥0 Zk+1R = fkR (2.24)
⇔ (H⊥0 − λ0)Zk+1R − (H⊥0 − λ0)P0Z
k+1R = fkR
⇔ (H⊥0 − λ0)Zk+1R + λ0P0Z
k+1R = fkR
⇔ (H0 − λ0)Zk+1R −H0P0Z
k+1R − P0H0Z
k+1R + P0H0P0Z
k+1R + λ0P0Z
k+1R = fkR
⇔ (H0 − λ0)Zk+1R − P0H0Z
k+1R + λ0P0Z
k+1R = fkR
⇔ (H0 − λ0)Zk+1R = fkR (2.25)
Note thatP⊥0 f
kR = fkR − P0f
kR = fkR − P0IRψ0 + εRψ0 = fkR.
Thus (2.25) is solvable, however the solution is not unique. Since Zk+1R decays exponen-
tially, we introduce the 6-dimensional box Ω := [−b, b]6 and solve (2.25) in the truncateddomain Ω with zero boundary conditions, i.e. we solve
(H0 − λ0)Zk+1R = fkR in H1
0 (Ω). (2.26)
In order to dene a suitable Finite Element space we dene the set of equidistant pointsin the interval [−b, b]
ξi = −b+ ih and h = 2b/(n+ 1), (2.27)
9
where n ∈ N is the grid size in one dimension. The corresponding space of piecewiselinear polynomials that are zero on the boundary is given by
S := ϕ ∈ C0([−b, b]) |ϕ[ξi,ξi+1] ∈ P1, 0 ≤ i ≤ n ∩H10 ([−b, b]).
We denote by bi, 1 ≤ i ≤ n the classical hat basis functions of S, i.e.,
bi(x) =
h−1(x− ξi−1) for x ∈ [ξi−1, ξi],
h−1(ξi+1 − x) for x ∈ [ξi, ξi+1].
A nite dimensional subspace of H10 (Ω) = H1
0 ([−b, b]6) is then given by the tensor product
S :=6⊗
k=1
S.
We dene the index setI := 1, . . . , n3
and use a bold typeface for elements in this set, e.g. I 3 i = (i1, i2, i3). Analogous to theone-dimensional case, basis functions for the six-dimensional space S are given by
bi,j(x, y) := bi(x)bj(y) :=
3∏k=1
bik(xk)
3∏l=1
bjl(yl), i, j ∈ I.
In order to solve (2.26) approximately with a Finite Element method we introduce thefollowing weak form of the problem: Find Zk+1
R,S ∈ S such that∫Ω
(H0 − λ0)Zk+1R,S (x, y)v(x, y)dxdy =
∫ΩfkR(x, y)v(x, y)dxdy
for all v ∈ S. Using the ansatz
Zk+1R,S (x, y) =
∑i,j∈I
ck+1i,j bi,j(x, y)
with unknown coecients ck+1i,j ∈ R this leads to: Find
ck+1i,j
i,j∈I
∈ R6n such that
∑i,j∈I
ck+1i,j ((H0 − λ0)bi,j, bk,l)L2(Ω) = (fkR, bk,l)L2(Ω) (2.28)
for all k, l ∈ I. The conventional way of solving (2.28) is to introduce a numeration of theunknowns, to set up the (sparse) Galerkin matrix and the right-hand side vector accordingto this numeration and to solve the arising linear system with an iterative scheme. Sincewe have a six-dimensional problem and the number of grid points in one direction n willbe at least several thousands the total number of unknowns can easily exceed 1018. Theconventional solution process is thus prohibitively expensive.In order to solve (2.28) we pursue the following strategy that involves low-rank tensorrepresentations, especially the so-called Tensor-Train format (see Section 3):
10
1. Represent the Galerkin matrix of the operator in the canonical tensor format. Sincethe operator is the sum of the Laplace operator, the identity operator and the oper-ator 1/|x|+ 1/|y| we will derive a tensor representation for each of these operators.
2. Represent the right-hand side in the canonical format or directly in the Tensor-Trainformat (TT-format) via a cross approximation scheme (see Section 3).
3. Convert the tensors in the canonical format numerically to the TT-format.
4. Solve the the linear system directly in the TT-format.
Before we discuss these steps in detail we give a brief introduction to recently developedlow-parametric tensor representations which will play a crucial role in our scheme.
3 Low-rank tensor representations
In this section we introduce the basic low-rank tensor representations that will be usedin the following. For a general introduction to low-parametric tensor formats we referto [18, 17]. For a detailed discussion of the Tensor-Train format which will be of specialimportance here we refer to [27].
A d-th order tensor A ∈ RI with I = I1 × · · · × Id, where Ij = 1, . . . , nd is a multidi-mensional array which can be represented entry-wise by
A = [A(i1, . . . , id)] = [Ai1,...,id ], with ij ∈ Ij , j = 1, · · · , d.
In the following we only consider equal-size tensors and therefore set nj = n and Ij =I := 1, . . . , n for j = 1, . . . , d. If all entries Ai1,...,id of A are stored explicitly, then Ais said to be in full format. The storage complexity of such a representation is nd andtherefore scales exponentially in the dimension d. The full format suers from the curseof dimensionality and can only be used for small dimensions, typically d ≤ 3, and smallmode sizes n.In order to relax (or even circumvent) the curse of dimensionality, dierent low-parametrictensor representations were introduced which often can drastically reduce the storage com-plexity compared to the full format. Furthermore these formats also allow the ecienthandling of operations like addition, matrix-vector multiplication or element-wise multi-plication which crucial for the solution of practical problems. The most commonly usedtensor formats include the canonical format, the Tucker format, the Tensor-Train (TT),the Quantics Tensor-Train (QTT) and the hierarchical format.In the following we will focus on the canonical format in combination with the TT- andQTT-format respectively.
Denition 3.1 (Canonical format). A tensor A is said to be in the canonical format if
it can be written as a linear combination of elementary tensors
A =r∑j=1
u(1)j ⊗ u
(2)j ⊗ · · · ⊗ u
(d)j , u
(i)j ∈ Rn,
where r is the so-called canonical rank, u(i)j are canonical factors and ⊗ denotes the
tensor-product for vectors.
11
If A is represented in the canonical format the storage complexity reduces to drn com-pared to nd for the full format. Furthermore basic operations involving tensors in thecanonical format can be performed with linear scaling in the dimension d.
The concept of a canonical representation of tensors induces the concept of a canonicalrepresentation of linear operators (see [17, 22]). This plays an important role in practicesince it often allows the low-parametric representation of matrices that result from FiniteElement approximations of PDEs (e.g. discretized Laplace operator). We adopt thenotation in [22] and use a bold, calligraphic font to denote operators.Let Vj and Wj , j = 1, . . . , d be vector spaces and let the corresponding tensor productspaces be dened as
V =d⊗j=1
Vj , W =d⊗j=1
Wj .
Furthermore let linear mappings Aj : Vj → Wj be given. The Kronecker product of themappings Aj is the linear mapping
A :=d⊗j=1
Aj : V→W, where A
d⊗j=1
vj
=
d⊗j=1
(Ajvj)
for elementary tensors⊗d
j=1 vj ∈ V. An example of an r-term canonical operator istherefore
A =r∑
k=1
d⊗j=1
A(k)j , (3.1)
where A(k)j , 1 ≤ j ≤ d, 1 ≤ k ≤ r are linear mappings. In the case Vj = Wj = RI the
mappings A(k)j are matrices. In this case the required storage size of the canonical format
(3.1) is only O(rdn2) compared to O(n2d) for the full format.
One of the main limitations of the canonical format is the diculty to approximate agiven tensor in the full format with a tensor in the canonical format. Although dierentalgorithms are available for this problem (e.g. [13]) they are not guaranteed to workeven if a low-rank canonical approximation of a given tensor is known to exists. Anotherdisadvantage is that there are no robust ways to approximate a given tensor A in thecanonical format with another tensor B in the canonical format which satises (for agiven ε > 0):
‖A−B‖F < ε‖B‖F and rank(B) < rank(A),
where ‖ · ‖F denotes the Frobenius norm. Such tensor rounding procedures are very im-portant since basic linear algebra operations like addition or element-wise multiplicationnecessarily lead to increased tensor ranks. If many of such operations are performed theranks can grow excessively. Therefore ecient reapproximation schemes are necessary toreduce the ranks while maintaining the accuracy.Despite these diculties the canonical format will play an important role in our ap-plication. It is often the case that tensors arise from functions (e.g. evaluation of amultidimensional function on a product grid) or operators that are given analytically. If
12
these functions can be accurately approximated by separable functions it is straightfor-ward to obtain a canonical representation of the corresponding tensor.
Due to the drawbacks of the canonical format we will also make use of the so-calledTensor-Train (TT) format ([28, 27]).
Denition 3.2 (TT format). For a given rank parameter r = (r0, . . . , rd), rk ∈ N, wherewe set r0 = rd = 1, the rank-r Tensor-Train format contains all elements A such that
Ai1,...,id = A(1)i1A
(2)i2· · ·A(d)
id, (3.2)
where each A(k)ik
are rk−1 × rk matrices. The A(k) can be considered as three-dimensional
arrays of size rk−1 × n× rk and are called the cores of the TT-tensor A.
The Tensor-Train representation reduces the storage complexity of a tensor in the fullformat to O(dr2n), where r = max0≤k≤d rk. All basic operations with TT-tensors canbe performed with linear complexity in d and polynomial complexity in r. An importantadvantage of the TT-format compared to the canonical format is that robust SVD-basedalgorithms are available in order to convert a tensor in the full format to the TT-format.Furthermore ecient tensor rounding methods are available (see [27] for details).In contrast to the canonical format the TT-format does not have one but d− 1 TT-ranks(recall that r0 = rd = 1). Since it is often convenient to characterize these ranks with asingle number we introduce the notion of the eective rank of a TT-tensor A. In the caseof equal mode sizes n it is dened as the positive solution of the quadratic equation
nr1 +d−1∑k=2
rk−1nrk + rd−1n = nr +d−1∑k=2
rnr + rn (3.3)
and will be denoted by re or erank(A). Note that the left-hand side of (3.3) characterizesthe memory requirement of the given TT-tensor. Dening re via (3.3) therefore makesthis rank eective with respect to memory.
As above we extend the denition of the Tensor-Train format to operators, in particularto matrices. Let the mappings A(k)
j in (3.1) be matrices. Then elements in A can benaturally index by 2d-tuples (i1, . . . , id, l1, . . . , ld), where (i1, . . . , id) represent rows and(l1, . . . , ld) columns of A. Analogous to Denition 3.2 the matrix A is said to be in theTT-format if each element can be represented as
A[i1, . . . , id, l1, . . . , ld] = A(1)i1,l1
A(2)i2,l2· · ·A(d)
id,ld, (3.4)
where A(k)ik,lk
are rk−1×rk matrices. The storage complexity of a matrix in the TT-formattherefore is of order O(dr2n2). The important operation of multiplying a TT-matrix in(3.4) with a TT-vector in (3.2) can be eciently performed with linear complexity in dand polynomial complexity in r.
As we will see later it will be crucial to convert canonical representations (or approxima-tions) of certain tensors to their corresponding TT-representation. We recall the followingtheoretical result:
13
Theorem 3.3 ([27]). If a tensor A admits a canonical approximation with r terms andaccuracy ε, then there exists a TT-approximation with TT-ranks rk < r and accuracy√d− 1ε.
As furthermore shown in [27] the numerical conversion from the canonical representationto the TT-format is simple and can be computed eciently.
In our application the parameter n will be the number of grid points in one dimensionof a 6-dimensional product grid that is used to approximate PDEs with piecewise linearpolynomials. In order to obtain accurate approximations, n is therefore typically ratherlarge. In the case of large mode sizes n the storage complexity of a TT-tensor can inmany cases be further reduced by using the so-called quantics-TT (QTT) approximationmethod ([30, 24]). The underlying idea of the QTT approximation is to further exploitthe linear scaling of storage and operations in the dimension d by rst reshaping a givend-dimensional tensor with mode size n to a log2 n · d-dimensional tensor with mode size2 and to approximate the resulting tensor in the TT-format. In this way the storagecomplexity of the approximation can be reduced from O(dnr2) for the TT-tensor withmode size n to O(2d log2 nr
2) for the corresponding QTT approximation. We refer to[22, 24] for a precise denition and applications of the QTT approximation.
It is often straightforward to nd approximate canonical representations of function-related tensors (typically by exploiting certain expansions of the underlying function). Inorder to obtain (approximate) TT-representations of such function-related tensors we willapply one of the following two strategies:
Obtain an approximation of the tensor in the canonical format and convert it nu-merically to the TT-format. As Theorem 3.3 indicates such a TT-representationalways exists and can be computed eciently.
Use a TT-cross approximation algorithm (explained below) to set up a TT-repre-sentation directly using only certain entries of the full tensor.
The rst strategy to obtain a TT-approximation of a given tensor is preferable because itcan typically be computed very eciently and furthermore the error of the approximationcan be rigorously estimated and controlled. Since it is however not always possible toobtain a canonical approximation of a function-related tensor we will also make use of(heuristic) cross approximation schemes for TT-tensors. These schemes only require thatthe tensor that has to be approximated is indeed of low rank and that individual entriesof this tensor can be computed (or approximated). The idea behind these algorithmsis to reconstruct a TT-approximation of a tensor with the knowledge of only a smallportion of all tensor entries. Cross approximation schemes were originally designed forthe matrix case ([3, 33]) and were recently generalized to dierent tensor formats ([29, 2]).We recall a theoretical result from [29] which states that a TT-tensor with ranks boundedby r can be recovered exactly by computing only O(dnr2) elements at certain positionsof this tensor. In the case where the given tensor is only approximately of low rankthe important question is how to choose the individual elements in such a way that theapproximation can be eciently and accurately computed. A detailed discussion of TT-cross approximation schemes (that will be frequently used in our algorithm) is beyondthe scope of this paper and we therefore refer to the literature for details.
14
The strategy discussed in Section 2.4 leads to a linear system of equations where theGalerkin matrix and the right-hand side are represented in the TT-format. Dierentsolvers for these kind of linear systems are available ([10, 20, 9]). Here we use alternatingminimal energy methods devloped in [11, 12], where the main idea is to seek the solutionof a linear system AX = B as minimizer of the energy function
J(X) = ‖X− X‖2A =(X,AX
)− 2R(X,B) + const,
where ‖U‖2A = (U,AU) and X denotes the exact solution. This (dicult) global mini-mization problem over all tensor trains X is then replaced by local minimization problemsover each core X(k) of X subsequently in a cycle. More specically we obtain the localproblems by assuming that all but the k-th core of the current TT-tensor X are xed andwe only minimize over the core X(k), i.e.,
X(k)new = arg min
X(k)J(X).
The new TT-tensor Xnew then has the cores X(1), . . . , X(k), X(k)new, X(k+1), . . . , X(d). It
can be shown that each local minimization problem is equivalent to a linear system ofmoderate size. For a detailed description of these methods we refer to the correspondingliterature.
4 Low-parametric representation of the involved tensor
4.1 Representation of the Galerkin matrix
Our aim in this section is to represent the left-hand side of (2.28) in the canonical tensorformat. In order to simplify the presentation we introduce the notation
AT :=
(Tbi,j, bk,l)L2(Ω)
i,k,j,l∈I , (4.1)
where T is an operator. In the following we derive ecient representations of AT for T ∈−∆, |x|−1 + |y|−1, Id in order to obtain a representation of AH0−λ0 which correspondsto the Galerkin matrix in (2.28).
4.1.1 Representation of A−∆
Recall that the three-dimensional Laplace operator applies to a separable function η(x) =η1(x1)η2(x2)η3(x3) as follows
∆xη(x) = η′′1(x1)η2(x2)η3(x3) + η1(x1)η′′2(x2)η3(x3) + η1(x1)η2(x2)η′′3(x3).
Analogous formulas hold true in higher dimensions. Let
∆ := ∆x + ∆y
15
be the six-dimensional Laplacian. Then the entries of A−∆ are of the form∫Ω−∆bi,j(x, y)bk,l(x, y)dxdy
=−∫
Ω∆xbi,j(x, y)bk,l(x, y)dxdy −
∫Ω
∆ybi,j(x, y)bk,l(x, y)dxdy
=
3∑q=1
(3∏
m=1
βqim,km
)(3∏
m=1
γjm,lm
)+
3∑q=1
(3∏
m=1
γim,km
)(3∏
m=1
βqjm,lm
), (4.2)
where
γim,km = (bim , bkm)L2(R), βqim,km =
(b′im , b
′km
)L2(R) if q = m
γim,km otherwise.
The Galerkin matrix A−∆ ∈ Rn6×n6corresponding to this operator in the tensor basis
bi,j can be represented in the canonical format as a tensor of rank 6 by
A−∆ = A(1) ⊗ S(2) ⊗ S(3) ⊗ S(4) ⊗ S(5) ⊗ S(6)
+ S(1) ⊗A(2) ⊗ S(3) ⊗ S(4) ⊗ S(5) ⊗ S(6)
+ S(1) ⊗ S(2) ⊗A(3) ⊗ S(4) ⊗ S(5) ⊗ S(6)
+ S(1) ⊗ S(2) ⊗ S(3) ⊗A(4) ⊗ S(5) ⊗ S(6)
+ S(1) ⊗ S(2) ⊗ S(3) ⊗ S(4) ⊗A(5) ⊗ S(6)
+ S(1) ⊗ S(2) ⊗ S(3) ⊗ S(4) ⊗ S(5) ⊗A(6),
where A(l), S(l) ∈ Rn×n, l = 1, . . . , 6 are given by
A(l) :=
(b′i, b′k)L2(R)
ni,k=1
=1
htridiag−1, 2,−1,
S(l) :=
(bi, bk)L2(R)
ni,k=1
=h
6tridiag1, 4, 1
and ⊗ denotes the Kronecker product for matrices ([18]). Recall that this canonicalrepresentation of A−∆ will be numerically transformed to the TT-format where ecientoperations and rounding procedures are available. Note that A−∆ might have an ex-plicit representation in the TT-format (for nite dierence discretizations of −∆ thiswas derived in [21]) which would make the setup of the canonical representation and theconversion superuous.
Remark 4.1. Note that in (4.2) the indices are organized in pairs (i1, k1), (i2, k2), . . . andthat this ordering carries over to the structure of A−∆. This organization of the indices
is crucial in order to obtain a low-rank representation of A−∆.
4.1.2 Representation of AId
Let S(l) be dened as above. Then the tensor representation of the discretized identityoperator has the form
AId = S(1) ⊗ S(2) ⊗ S(3) ⊗ S(4) ⊗ S(5) ⊗ S(6).
16
4.1.3 Representation of A|x|−1+|y|−1
In order to represent the matrix A|x|−1+|y|−1 = A|x|−1 +A|y|−1 in a compressed tensor for-mat we have to approximate the kernel functions |x|−1 and |y|−1 with separable functions(see [4]). We start by expressing |x|−1 in terms of its inverse Laplace transform
1
|x|=
1√π
∫R
e−|x|2t2dt =
1√π
∫R
3∏ν=1
e−x2νt
2dt, for |x| > 0. (4.3)
Since entries of A|x|−1 are of the form
A|x|−1 [i, j,k, l] =
(1
|x|bi(x), bk(x)
)L2([−b,b]3)
(bj(y), bl(y))L2([−b,b]3) (4.4)
and the second factor can be easily separated as in Section 4.1.1 we focus on the rstfactor and note that with Fubini's theorem and (4.3) we have(
1
|x|bi(x), bk(x)
)L2([−b,b]3)
=1√π
∫R
∫[−b,b]3
3∏ν=1
e−x2νt
2bi(x)bk(x)dxdt
=
∫R
3∏ν=1
G(ν)iν ,kν
(t)dt
with
G(ν)iν ,kν
(t) = π−1/6
∫ b
−bbiν (xν)bkν (xν)e−x
2νt
2dxν . (4.5)
The tensor we have to approximate is therefore given by
G =
∫R
3⊗ν=1
G(ν)(t)dt with G(ν)(t) ∈ Rn×n.
An accurate way to approximate this integral of a tensor valued function is based on sincquadrature ([4, 18]), i.e,
G ≈ GM :=M∑
µ=−Mgµ
3⊗ν=1
G(ν)(tµ)
withtµ = µhM , gµ = hM , hM = C0 log(M)/M,
where C0 > 0. This type sinc quadrature approximation typically leads to exponentialconvergence in M , i.e,
‖G − GM‖ ≤ C e−α√M ‖G‖,
where C,α ∈ R+ and ‖ · ‖ denotes the Frobenius norm. Here we use an alternative sincquadrature. We rst use the variable transformation t = ζ−1 sinh(ζz), where ζ ∈ R+ is asuitably chosen parameter and then apply sinc quadrature to the transformed integrand,i.e.,
G =
∫R
cosh(ζz)
3⊗ν=1
G(ν)(ζ−1 sinh(ζz))dz ≈M∑µ=0
gµ
3⊗ν=1
G(ν)(tµ) =: GM , (4.6)
17
where the quadrature nodes and weights are given by
tµ = ζ−1 sinh(µζhM ), gµ =
hM for µ = 0,
2hM cosh(µζhM ) for 0 < µ ≤M(4.7)
and hM = C0 log(M)/M is as above. Note that we took advantage of the symmetry ofthe integrand which reduces the number of function evaluations from 2M + 1 to M + 1 in(4.7). Furthermore this sinc quadrature converges asymptotically with an improved rate.It can be shown that
‖G − GM‖ ≤ C e−αM/ log(M) ‖G‖,
where again C,α ∈ R+. The reason for the improved convergence rate is the fasterdecay of the integrand due to the variable transformation. In the following we want tonumerically determine optimal values for ζ, C0 and M (see also [4] for the case ζ = 1).The choice of these parameters is crucial for the performance of the sinc quadrature sincethey have to be chosen such that each entry of G, i.e.,
G[i,k] =
∫R
cosh(ζz)3∏
ν=1
G(ν)iν ,kν
(ζ−1 sinh(ζz))dz (4.8)
is approximated accurately. More precisely only one sinc quadrature (one set of nodesand weights) must be capable of approximating the whole tensor G accurately. This isonly possible if the integrands in (4.8) have a similar shape for each set of indices. This is,however, not the case in general but can be controlled to some extend with the parameterζ. To illustrate this we consider the case b = 10 and n = 200 and plot the integrand in(4.8) for two dierent tensor entries and ζ = 1 and ζ = 15 respectively (see Figure 4.1).
−8 −6 −4 −2 0 2 4 6 8
10−10
10−8
10−6
10−4
10−2
z
i=k=[1 1 1]
i=k=[100 100 100]
(a) ζ = 1
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
10−10
10−8
10−6
10−4
10−2
100
z
i=[100 100 100],k=[100 100 100]i=[1 1 1]. k=[1 1 1]
(b) ζ = 15
Figure 4.1: Plot of the integrands in (4.8) for dierent tensor entries and parameters ζ. we setb = 10 and n = 200.
It becomes evident that the decay of the integrands strongly depends on the positionof the corresponding entry in the tensor. Entries whose basis functions are supportedclose to zero decay quickly only for larger values of z. Due to the dierent shapes of theintegrand it is problematic to choose parameters C0 and M such that the resulting sinc
18
quadrature is accurate for every tensor entry (unless M is chosen very large). In orderto relax this issue we choose ζ suciently large such that the slowly decaying integrandsare damped stronger and the quickly decaying integrands remain basically unaected. InFigure 4.1(b) the shape of the integrands for ζ = 15 is illustrated.
−10 −5 0 5 1010
−11
10−10
10−9
10−8
10−7
10−6
z
i=k=[1 1 1]
i=k=[104 104 104]
(a) ζ = 1
−0.4 −0.2 0 0.2 0.4 0.610
−11
10−10
10−9
10−8
10−7
10−6
10−5
z
i=k=[1 1 1]
i=k=[104 104 104]
(b) ζ = 30
Figure 4.2: Integrands for dierent tensor entries where b = 10 and n = 20000.
If we increase the number of grid points to n = 20000 this issue becomes even more severe(cf. Figure 4.2 where we plot again the occurring integrands at dierent positions of thetensor). In this case we have to increase the value of ζ to 30 in order to compensate thefast decay of certain integrands (cf. Figure 4.2(b)).In Figure 4.3 we illustrate the relative error of the sinc quadrature (4.6) for n = 200 andn = 20000 for dierent tensor entries (4.8). As discussed above we set ζ = 15 and ζ = 30to improve the accuracy of the approximation across all tensor entries. The constant C0
which is present in the denition of hM controls the stepsize of the quadrature rule. Smallvalues for C0 lead to more accurate results for quickly decaying integrands while largervalues are better for more slowly decaying integrands. In practice C0 has to be chosen suchthat all occurring integrals are approximated with similar accuracy. In our applicationC0 ∈ [0.1, 0.2] (depending on n) lead to the best results. As shown in Figure 4.3 we obtainexponential convergence of the sinc quadrature rule with respect to M . Furthermore theerrors are of comparable magnitude due to the calibration of the parameters ζ, C0 andM .
With (4.4) and (4.6) the Galerkin matrix associated with the operator |x|−1 can be(approximately) represented by
A|x|−1 =
(1
|x|bi(x), bk(x)
)L2([−b,b]3)
(bj(y), bl(y))L2([−b,b]3)
i,k,j,l∈I
≈M∑µ=0
gk ·G(1)(tk)⊗G(2)(tk)⊗G(3)(tk)⊗ S(4) ⊗ S(5) ⊗ S(6) =: AM|x|−1 .
Similarly we get for the Galerkin matrix associated with the operator |y|−1 the represen-
19
0 10 20 30 40 50 60
10−12
10−10
10−8
10−6
10−4
10−2
100
M
rel.
erro
r
i=k=[1 1 1]i=k=[100 100 100]i=[1 151 199]k=[1 150 200]
(a) C0 = 0.2, b = 10, n = 200.
0 20 40 60 80 10010
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
M
rel.
erro
r
i=k=[1 1 1]
i=k=[104 104 104
i=[1 10001 19999]k=[1 10000 20000]
(b) C0 = 0.14, b = 10, n = 20000.
Figure 4.3: Relative errors of the sinc quadrature (4.6) for dierent tensor entries and grid sizesn.
tation
A|y|−1 ≈M∑µ=0
gk · S(1) ⊗ S(2) ⊗ S(3) ⊗G(4)(tk)⊗G(5)(tk)⊗G(6)(tk) =: AM|y|−1 .
Finally we obtain the approximate representation
A|x|−1+|y|−1 ≈ AM|x|−1+|y|−1 := AM
|x|−1 + AM|y|−1 .
Remark 4.2. AM|x|−1 and AM
|y|−1 are canonical tensors of rank M + 1. In practice M can
be chosen (depending on the desired accuracy) in the range 30− 50.
4.1.4 Representation of AH0−λ∞
The results of the previous subsections show that the Galerkin matrix associated withproblem (2.28) can be approximately represented by
AH0−λ∞ ≈ AMH0−λ∞ :=
1
2A−∆ −AM
|x|−1+|y|−1 + AId.
4.2 Representation of the right-hand side
In this section we represent the tensor on the right-hand side of (2.28) in a low-ranktensor format. Recall that
(fkR, bk,l)L2(Ω) =(
(λkR − λ∞)Y kR − P⊥0 IRY k
R − IRψ0 + εRψ0, bk,l
)L2(Ω)
. (4.9)
Note that Y kR is the projected Galerkin solution of a previous iteration of (2.25), i.e.,
Y kR = P⊥0 Z
kR,S =
∑i,j∈I
cki,jP⊥0 bi,j(x, y) =
∑i,j∈I
cki,jbi,j(x, y)−∑i,j∈I
cki,j (ψR, bi,j)ψR(x, y).
(4.10)
20
The coecients tensor cki,j will be available in the TT-format due to the computationsin the previous iteration. In the following we represent all the components in a low-ranktensor format that are necessary to obtain a representation of (4.9). In order to simplifythe presentation we introduce the following notations:
Bg :=
(g, bk,l)L2(Ω)
k,l∈I (4.11)
Ck :=ckk,l
k,l∈I
. (4.12)
4.2.1 Representation of Bψ0
Following [23] we can express the exponential function e−√ρ for ρ ≥ 0 via its Laplace
transform as
e−√ρ =
1
2√π
∫R+
τ−3/2e−14τ−τρdτ =
1
2√π
∫R
e−τ2− 1
4e−τ−ρeτdτ.
Therefore we have∫R3
e−|x|bi(x)dx =
∫R3
e−|x|bi1(y1)bi2(y2)bi3(y3)dx
=1
2√π
∫R3
∫R
e−τ2− 1
4e−τ−(x21+x22+x23)eτ bi1(x1)bi2(x2)bi3(x3)dτdx
=1
2√π
∫R
e−τ2− 1
4e−τ∫R3
e−(x21+x22+x23)eτ bi1(x1)bi2(x2)bi3(x3)dxdτ
=1
2√π
∫Rβ(τ)ηi1(τ)ηi2(τ)ηi3(τ)dτ, (4.13)
withβ(τ) := e−
τ2− 1
4e−τ and ηi(τ) :=
∫R
e−x2eτ bi(x)dx.
The function ηi can be expressed in closed form in terms of error functions. For simplicityhowever we compute approximations of these functions with standard Gauss-Legendrequadrature. In order to approximate (4.13) we proceed as in (4.6) and apply a sincquadrature after a suitable variable transformation τ = sinh(z), i.e.,
1
2√π
∫Rβ(τ)ηi1(τ)ηi2(τ)ηi3(τ)dτ ≈ 1
2√π
M∑µ=−M
gµβ(tµ)ηi1(tµ)ηi2(tµ)ηi3(tµ). (4.14)
withgµ = hM cosh(µhM ), tµ = sinh(µhM ), hM = C0 log(M)/M.
Note that the parameter ζ in (4.7) is set to 1 here. In contrast to above the accuracy ofthe sinc quadrature could not be signicantly increased by choosing ζ otherwise.Figure 4.4 illustrates the shape of the integrands in (4.13) after the substitution τ =sinh(z) for dierent parameters and indices. In contrast to Section 4.1.3 these functionsare not symmetric and the dierences in the decay behavior (depending on the tensorentry) is not as substantial as before. Figure 4.5 shows the relative error of the approxi-mation (4.14) for dierent values of M and dierent tensor entries. The calibration of C0
21
−3 −2 −1 0 1 2 3 40
0.2
0.4
0.6
0.8
1
1.2
1.4x 10−3
z
i=[100,100,100]i=[120,100,100]i=[100,120,80]
(a) n = 200
−3 −2 −1 0 1 2 3 40
0.2
0.4
0.6
0.8
1
1.2
1.4x 10−9
z
i=[10000, 10000, 10000]i=[9000, 10000, 10000]i=[9000, 10000, 11000]
(b) n = 20000
Figure 4.4: Integrands in (4.13) after the substitution τ = sinh(z) for dierent tensor entrieswhere b = 10.
0 10 20 30 40 50 6010−12
10−10
10−8
10−6
10−4
10−2
100
M
i=[100, 100, 100]i=[120, 100, 100]i=[100, 120, 80]
(a) C0 = 1.1, b = 10, n = 200
0 10 20 30 40 50 6010−12
10−10
10−8
10−6
10−4
10−2
100
M
i=[10000, 10000, 10000]i=[9000, 10000, 10000]i=[9000,10000,11000]
(b) C0 = 1.1, b = 10, n = 20000
Figure 4.5: Relative quadrature errors for dierent tensor entries and grid sizes.
in this case is less critical than in the preceding section. A value that leads to accurateresults for dierent grid sizes and tensor entries is C0 = 1.1.The tensor Bψ0 can therefore be approximately represented by
Bψ0 =(
e−|x|, bi
)i∈I⊗(
e−|y|, bj
)j∈I
≈ c2
4π2
M∑µ=−M
gµβ(tµ) ·3⊗i=1
η(tµ)
⊗( M∑ν=−M
gνβ(tν) ·3⊗i=1
η(tν)
)=: BM
ψ0,
whereη(τ) = (ηi(τ))ni=1 .
22
Note that this a tensor has canonical rank (M+1)2. In Table 1 we show that the TT-ranksare signicantly smaller in practice. We choose the mode size 2 for the TT-tensors (whichrequires that the grid size n is a multiple of 2), i.e., we compute the QTT approximationwith an accuracy of 10−8 and denote the resulting tensor BM,QTT
ψ0.
M\n 27 28 29 210 211 212
10 25.9 30.0 32.1 33.1 33.6 33.8
20 26.9 31.6 35.3 37.8 39.6 40.8
30 26.8 31.3 35.1 37.7 39.9 41.1
Table 1: Eective QTT-ranks as dened in (3.3) of BM,QTTψ0
for dierent accuracies of the sincquadrature (controlled by M) and dierent grid sizes n.
4.2.2 Representation of BIRψ0
In order to compute a low-rank approximation of
BIRψ0 =
∫Ω
(− 1
|x− 2Re|− 1
|y + 2Re|+
1
|x− y − 2Re|+
1
2R
)ψ0bi,j dxdy
i,j∈I
we can not use the same techniques as in the previous sections since the kernel functionIRψ0 does not admit a simple and accurate, analytic expansion in terms of separablefunctions. Instead we use a cross approximation scheme that was introduced in Section 3to compute a QTT-approximation of this tensor, denoted by BQTT
IRψ0, directly. This scheme
only requires the ability to compute single tensor entries which are six-dimensional inte-grals in our case. We approximate these integrals by tensorized Gauss-Legendre quadra-ture. Since the computation of the required entries in BIRψ0 can be parallelized thecomputational time remains moderate.
n = 29 n = 210 n = 211
re accuracy re accuracy re accuracyR = 5, b = 5 81.2 1.0 · 10−1 68.2 1.8 · 10−2 66.0 5.0 · 10−2
R = 5, b = 7 106.9 2.5 · 10−1 107.3 3.0 · 10−1 108.1 4.0 · 10−1
R = 10, b = 10 62.5 1.8 · 10−4 62.7 1.6 · 10−4 60.2 2.7 · 10−4
R = 15, b = 15 58.2 6.2 · 10−5 55.3 1.1 · 10−4 54.8 1.1 · 10−4
R = 15, b = 10 56.6 8.2 · 10−5 57.5 1.0 · 10−4 56.6 1.1 · 10−4
Table 2: Eective QTT-ranks as dened in (3.3) of BQTTIRψ0
for dierent values of R, b and n.
In Table 2 we list the eective ranks and accuracies of BQTTIRψ0
for dierent values of R, band grid sizes n. Since the cross approximation algorithm is a heuristic procedure we cannot measure the actual error of the approximation. By `accuracy' we therefore denotethe dierence (measured in the Frobenius norm) of two consecutive iterations within thecross approximation scheme.Table 2 indicates that BQTT
IRψ0can be accurately approximated with moderate ranks if R
is suciently large (typically R > 7). For smaller R and b = R this tensor becomes
23
more dicult to approximate since the damping due to ψ0 is less strong. Note that forb > R the singularity due to |x − y − 2Re|−1 enters the computational domain whichhas a negative impact on the accuracy on the cross approximation scheme and leads to asignicant rank growth.
Remark 4.3. Since
BIRψ0 = −B|x−2Re|−1ψ0−B|y+2Re|−1ψ0
+ B|x−y−2Re|−1ψ0+ B(2R)−1ψ0
(4.15)
it is in principle also possible to obtain a QTT-approximation of BIRψ0 by computing ap-
proximations of the individual tensors on the right-hand side using a combination of ana-
lytic expansions and cross approximation and to sum up the results. In practice however
this leads to inaccurate approximations of BIRψ0 due to numerical cancellation eects.
4.2.3 Computation of λkR − λ0
With (2.14) and the denition of Y kR we get
λkR − λ0 = εR +(IRψR, Y
kR
)=∑k,l∈I
ckk,l (IRψR, bk,l) + εR
1−∑k,l∈I
ckk,l (ψR, bk,l)
.
With the results from the preceding sections we obtain the approximation
λkR − λ0 = 〈Ck,BIRψ0〉+ εR
(1− 〈Ck,Bψ0〉
)=: dkR, (4.16)
where 〈·, ·〉 denotes the scalar product. An approximation of λkR − λ0 can be obtained by
λkR − λ0 ≈ 〈Ck,BQTTIRψ0〉+ εR
(1− 〈Ck,BQTT
ψ0〉)
=: dk,QTTR .
4.2.4 Representation of BY kR
It holds
BY kR=
∑
k,l∈Icki,jbi,j −
∑i,j∈I
cki,j (ψR, bi,j)ψR, bk,l
k,l∈I
=
∑
i,j∈Icki,jbi,j, bk,l
k,l∈I
− 〈Ck,BψR〉BψR .
Since ∑i,j∈I
ci,jbi,j, bk,l
=∑
ν,ξ∈−1,0,13ck+ν,l+ξ (bk+ν,l+ξ, bk,l) ,
and for k + ν, l + ξ ∈ I we have
(bk+ν,l+ξ, bk,l) = θν · θξ, where θν :=
(h
6
)3
43−∑3i=1 |νi|, (4.17)
24
we obtain ∑
i,j∈Ici,jbi,j, bk,l
k,l∈I
=∑
ν,ξ∈−1,0,13θνθξ ·Ck
ν,ξ.
Here Ckν,ξ are shifted versions of the solution tensor Ck. They are dened by
Ckν,ξ := ck+ν,l+ξk,l∈I , ν, ξ ∈ −1, 0, 13 ,
where we formally set ckk+ν,l+ξ = 0 for k + ν, l + ξ 6∈ I.We obtain a QTT approximation BQTT
Y kRof BY kR
by replacing all the involved tensors above
by their respective QTT approximations.
4.2.5 Representation of BIRYkR
In order to compute a QTT approximation of
BIRYkR
=(IRY
kR , bk,l
)k,l∈I
(4.18)
we use the cross approximation algorithm as in Section 4.2.2. The evaluation of singletensor entries within this scheme could be done as before with tensorized Gauss-Legendrequadrature. However, since Y k
R does not have a simple analytic expression as has to beevaluated at the quadrature points via (4.10) this strategy is too expensive especiallysince BIRY
kRdepends on k and has to be computed multiple times within one simulation.
We therefore approximate the entries in (4.18) with a six-dimensional trapezoidal rule,i.e.,
BIRYkR
[k, l] ≈ h6 · IR(ξk, ξl) · Y kR(ξk, ξl) = h6 · IR(ξk, ξl) ·Yk
R[k, l], (4.19)
whereYkR = Ck − 〈Ck,BψR〉ψ0, ψ0 = ψ0(ξk, ξl)k,l∈I (4.20)
and ξl = (ξk1 , ξk2 , ξk3) is a vector of grid points (2.27). Thus, once a QTT approximationof Yk
R has been computed via (4.20) the tensor entry BIRYkR
[k, l] can be approximatedby evaluating the function IR at a six-dimensional grid point and computing the tensorentry Yk,QTT
R [k, l], which requires O(2 · log2(n) · 6r2) operations. The cross approxima-tion scheme in combination with the trapezoidal rule (4.19) lead to suciently accurateapproximations of BIRY
kRin all tested cases.
Remark 4.4. Equation (4.19) shows that
BIRYkR≈ h6 · IR Yk
R, with IR = IR(ξk, ξl)k,l∈I , (4.21)
where denotes the Hadamard product (element-wise multiplication). If we had a QTT
approximation of IR and YkR we could therefore compute BQTT
IRYkR
without using cross ap-
proximation. Unfortunately IR does not have an accurate low-rank QTT approximation
unless R is signicantly larger than b (recall Ω = [−b, b]6) and therefore the singularity
due to |x − y − 2R|−1 is suciently far away from the computational domain. Since we
typically set R = b the tensor BQTTIRY
kR
can not be eciently approximated via (4.21).
25
4.2.6 Representation of BP0IRYkRand BP⊥0 IRY
kR
In order to represent (4.9) in a low-rank format we need to compute BP⊥0 IRYkR. We rst
note that
BP0IRYkR
=(P0IRY
kR , bk,l
)k,l∈I
=(IRY
kR , ψR
)·Bψ0 .
The scalar can be eciently computed using
(IRY
kR , ψR
)=
IR∑
i,j∈Icki,jbi,j −
∑i,j∈I
cki,j (ψR, bi,j)ψR
, ψR
=∑i,j∈I
cki,j (IRbi,j, ψR)−∑i,j∈I
cki,j (ψR, bi,j) (IRψR, ψR)
= 〈Ck,BIRψR〉 − εR · 〈Ck,BψR〉.
Wit the denition of P⊥0 we get
BP⊥0 IRYkR
= BIRYkR−BP0IRY
kR.
As before we obtain the corresponding QTT representation by replacing all involvedtensors with their respective QTT approximations.
4.2.7 Representation of BfkR
We can nally compute the tensor with entries (4.9) which corresponds to the right-handside of our problem as
BfkR= dkR ·BY kR
−BP⊥0 IRYkR−BIRψ0 + εR ·Bψ0 ,
where dkR is as in (4.16). An approximation of this tensor in the QTT format is given by
BQTT
fkR= dk,QTTR ·BQTT
Y kR−BQTT
P⊥0 IRYkR
−BQTTIRψ0
+ εR ·BQTTψ0
.
4.2.8 Representation of B−R−3Bψ0
In order to initialize the iteration we have to represent B−R−3Bψ0in the QTT format.
Recall thatB(x, y) = x · y − 3(x · e)(y · e).
The tensorB−R−3Bψ0
=(−R−3Bψ0, bk,l
)k,l∈I
is computed via the cross approximation scheme mentioned above.
26
5 Numerical experiments
In this section we present the results of numerical experiments which show the perfor-mance and indicate the accuracy of the method for dierent separation distances R.All computations were done in MATLAB using the TT-toolbox 2.2 by I. Oseledets(http://spring.inm.ras.ru/osel/) on 2 × Intel Xeon E5-2640 processors (each ofwhich has 6 cores and 12 threads) with 256 GB RAM. The TT-toolbox allows the basicmanipulation of tensors in the (Q)TT-format and provides the TT-cross approximationalgorithm as well as the solver for the arising linear systems.
5.1 Verication of the method for equations of type (2.26)
We rst verify the validity of the solver for PDEs of type (2.26) using a Finite Elementscheme as described above. Therefore we consider the simple test problem
(H0 − λ0)Z =
(4− 1
|x|− 1
|y|
)g(x, y) in H1
0 ([−π, π]6), (5.1)
where
g(x, y) =3∏i=1
sin(xi)3∏j=1
sin(yj).
The exact solution of this problem is given by Z = g. We compute an approximaterepresentation of the Finite Element matrix in the QTT format as described in the pre-vious section. A representation of the corresponding right-hand side tensor is computedwith the cross approximation algorithm. The arising linear system is solved using alter-nating minimal energy methods as developed in [12]. Figure 5.1 shows the slice of theapproximate solution Znapprox
(·, π2 ,
π2 , ·,
π2 ,
π2
)and the corresponding pointwise error for
n = 210.
(a) Solution (b) Error
Figure 5.1: Approximation Zapprox(·, π2 ,
π2 , ·,
π2 ,
π2
)and pointwise error of (5.1) for n = 210.
Table 3 shows the relative error
errn :=‖Z − Znapprox‖L2([−π,π]6)
‖Z‖L2([−π,π]6)
27
of piecewise linear approximations for dierent grid sizes. The relative accuracy of theinvolved (approximate) tensor operations was set to 10−5. The table shows that the errordecreases by a factor 4 if the number of grid points is doubled. The cross approximationof the right-hand side and the solver of the linear system work as expected in all testedcases. Note that the exact solution Z can be represented exactly in the QTT-format withmaximal rank 2. The eective rank of the approximation Znapprox is not larger than 4.Depending on n the computations took 50s-150s.
# d.o.f. errn errn/errn+1 Eective rank of Znapproxn = 26 ≈ 6.8 · 1010 3.9 · 10−3 3.96 3.8
n = 27 ≈ 4.3 · 1012 1.0 · 10−3 3.96 3.2
n = 28 ≈ 2.8 · 1014 2.5 · 10−4 3.99 2.2
n = 29 ≈ 1.8 · 1016 6.3 · 10−5 4.23 2.8
n = 210 ≈ 1.2 · 1018 1.5 · 10−5 3.6
Table 3: Relative errors for piecewise linear approximations to (5.1).
5.2 Results for xed R.
In this section we set R = 10 and consider the computational domain Ω = [−10, 10]6, i.e.,b = 10. We solve
(H0 − λ0)ZkR = fk−1R in H1
0 (Ω)
λkR − λ0 = εR +(IRψ0, Y
k−1R
).
for k ∈ N and initial guess (2.15) for dierent grid sizes n. Recall that once an approx-imation of ZkR is available we obtain an approximation of the quantity of interest Y k
R
via the formula Y kR = P⊥0 Z
kR. The relative accuracy of all involved (approximate) ten-
sor operations, e.g., rounding, cross approximation, Hadamard product, is set to 10−4.Furthermore the accuracy of the solver for the arising linear system is also set to 10−4.
n = 29 n = 210 n = 211 n = 212
k = 0 −8.1226 · 10−7 −8.1231 · 10−7 −8.1233 · 10−7 −8.1231 · 10−7
k = 1 −8.1227 · 10−7 −8.1231 · 10−7 −8.1233 · 10−7 −8.1231 · 10−7
k = 2 −1.0665 · 10−7 −1.0664 · 10−7 −1.0666 · 10−7 −1.0666 · 10−7
k = 3 −1.0672 · 10−7 −1.0673 · 10−7 −1.0673 · 10−7 −1.0673 · 10−7
k = 4 −1.0672 · 10−7 −1.0672 · 10−7 −1.0673 · 10−7 −1.0673 · 10−7
k = 5 −1.0672 · 10−7 −1.0673 · 10−7 −1.0673 · 10−7 −1.0673 · 10−7
k = 6 −1.0672 · 10−7 −1.0673 · 10−7 −1.0673 · 10−7 −1.0673 · 10−7
k = 7 −1.0672 · 10−7 −1.0673 · 10−7 −1.0673 · 10−7 −1.0673 · 10−7
Table 4: λkR − λ0 for dierent grid sizes n.
Table 4 shows the convergence of the eigenvalue λkR for k = 0, . . . 7 and dierent grid sizesn. We can observe that the convergence is very fast with respect to k. Only four iterationsare sucient to reach the relative target accuracy of 10−4. The obtained value of λkR−λ0
28
for k = 7 and n = 212 is −1.0673 · 10−7. This compares to a value of −1.0652 · 10−7 givenby the approximation (2.7).
Eective rank of Yk,QTTR Maximal rank of Yk,QTT
R ‖Y kR‖2
k = 0 32.6 57 2.7198 · 10−3
k = 1 45.9 87 3.4703 · 10−4
k = 2 44.3 85 3.4721 · 10−4
k = 3 44.1 84 3.4719 · 10−4
k = 4 44.2 84 3.4719 · 10−4
k = 5 44.2 84 3.4719 · 10−4
Table 5: Properties of Y kR and Yk,QTTR for n = 212.
Table 5 shows dierent properties of the solution Y kR and its QTT approximation Yk,QTT
R
for n = 212. As before we can observe that Y kR converges quickly with respect to k.
Furthermore the ranks of the corresponding QTT tensors are moderate.
Recall that Y kR is a function of six variables x1, x2, x3, y1, y2, y3. In Figures 5.2 we plot a
slice of this solution by setting four of these variables to 0. As predicted by the theory wesee that Y k
R decays very quickly as (x, y) moves away from the origin. This is a crucialproperty for the method presented here since otherwise some of the involved tensors, e.g.,BIRY
kRin Section 4.2.5, could not be computed eciently and accurately due to large
ranks.
(a) View from side (b) View from above
Figure 5.2: Visualization of Y 6R (x1, 0, 0, y1, 0, 0) for n = 212 and R = 10.
5.3 Results for dierent separation distances
In this section we x the number of grid points to n = 212 and vary the separation distanceR. In all experiments we use 6 iterations within our iterative scheme. The accuracy ofall involved tensor operations is again set to 10−4. For the cross approximation wefurthermore limit the maximal rank of the approximate tensor to 120. This is necessarysince some of required tensors are dicult to approximate for smaller values of R, i.e.
29
5 ≤ R ≤ 7. In these cases the cross approximation algorithm then usually terminatesbefore the target accuracy of 10−4 is reached.In Figure 5.3 we plot |λR−λ0| for dierent values of R and compare our results with (2.7)and computed values obtained in [34]. We can observe that the computed ground statesare reasonable for all tested cases. For larger values of R they are almost indistinguishablefrom the approximation (2.7).
5 6 7 8 9 10 11 12 1310
−8
10−7
10−6
10−5
R
|λ
R−λ
0| with our method
analytic result|λ
R−λ
0| by L. Wolniewicz
Figure 5.3: Plot of |λR − λ0| for dierent R. The dashed red line represents the approximation(2.7). The black circles are the values obtained in [34].
In Table 6 we list the interaction energies λR − λ0 computed by our scheme for dierentvalues of R. Furthermore we show the eective ranks of the corresponding ground-statewavefunctionsY6,QTT
R . The accuracy of the computed ground states is dicult to estimatedue to the (heuristic) cross approximation scheme that is used by our method. In generalthe method is more accurate for larger R. While we do not expect a very high accuracyfor 5.0 ≤ R ≤ 7.0 due to rather high ranks of some of the involved tensors in this case(see Section 4.2.2), the ground states are computed accurately for R > 7. The ranks ofY6,QTTR are moderate in all tested cases. The computing times were approximately 12
hours for each value of R. Note that this depends signicantly on the number of iterationsand the number of quadrature points used in the cross approximation scheme to set uptensors like BQTT
IRψ0. Here we were rather conservative and used 6 quadrature points in each
direction, i.e. 66 in total, to compute single entries of BQTTIRψ0
. Another time consumingcomputation is the approximation of BIRY
kRas discussed in Section 4.2.5 since it has to
be done in each iteration and the TT-tensor Yk,QTTR has to be evaluated several times
within the cross approximation scheme. Here, we expect that an ecient implementationin a low-level programming language can signicantly reduce the computing times.
30
λR − λ0 Eective rank of Y6,QTTR
R = 5.0 −8.9520 · 10−6 27.7
R = 5.5 −4.1293 · 10−6 20.0
R = 6.0 −2.3335 · 10−6 49.5
R = 6.5 −1.4229 · 10−6 39.3
R = 7.0 −9.5037 · 10−6 37.4
R = 7.5 −6.2285 · 10−6 34.8
R = 8.0 −4.1749 · 10−6 20.8
R = 8.5 −2.8856 · 10−6 39.9
R = 9.0 −2.0325 · 10−6 39.2
R = 9.5 −1.4600 · 10−6 36.9
R = 10.0 −1.0673 · 10−6 38.7
R = 10.5 −7.9265 · 10−6 42.4
R = 11.0 −5.9265 · 10−6 40.4
R = 11.5 −4.5570 · 10−6 42.3
R = 12.0 −3.5191 · 10−6 39.7
R = 12.5 −2.7470 · 10−6 41.9
R = 13.0 −2.1657 · 10−6 40.7
Table 6: Results for dierent values of R.
6 Conclusion
In this paper we considered the numerical solution of the Born-Oppenheimer approxi-mation of the Schrödinger equation for two hydrogen atoms separated by a distance 2R.We showed that in case of larger separation distances the Feschbach-Schur perturbationmethod can be eectively used to derive a simpler problem with an operator that is inde-pendent of R and which can be solved using a simple iterative scheme. Low-rank tensormethods can be eciently and accurately applied to the arising six-dimensional PDEs ifthe separation distance is suciently large, i.e. R > 7. In this case the important crossapproximation algorithm and the employed solver for the linear systems work as expectedand deliver reliable results due to the low ranks of the involved tensors. For smaller valuesof R the tensor ranks grow which renders these methods less eective. However we stillget reasonable results for 5 ≤ R ≤ 7.Our method is currently implemented in MATLAB and uses the TT-toolbox by I. Os-eledets. We expect that an ecient (possibly parallel) implementation of our scheme ina low-level programming language can signicantly reduce the computational time.
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