Licensed Unlicensed Requires Authentication Published by De Gruyter May 31, 2017

Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach

Venera Khoromskaia and Boris N. Khoromskij
From the journal Computational Methods in Applied Mathematics
https://doi.org/10.1515/cmam-2017-0004

Abstract

This paper introduces and analyzes the new grid-based tensor approach to approximate solutions of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree–Fock equation over a spatial L1×L2×L3 lattice for both periodic and non-periodic problem setting, discretized in the localized Gaussian-type orbitals basis. In the periodic case, the Galerkin system matrix obeys a three-level block-circulant structure that allows the FFT-based diagonalization, while for the finite extended systems in a box (Dirichlet boundary conditions) we arrive at the perturbed block-Toeplitz representation providing fast matrix-vector multiplication and low storage size. The proposed grid-based tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) in the periodic case the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems in a box with Dirichlet boundary conditions are treated numerically by our previous tensor solver for single molecules, which makes possible calculations on rather large L1×L2×L3 lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic L×1×1 lattice chain in a 3D rectangular “tube” with L up to several hundred confirm the theoretical complexity bounds for the block-structured eigenvalue solvers in the limit of large L.

Keywords: Tensor Structured Numerical Methods for PDEs; 3D Grid-Based Tensor Approximation, Hartree–Fock Equation; Linearized Fock Operator; Periodic Systems; Lattice Sum of Potentials; Block Circulant/Toeplitz Matrix; Fast Fourier Transform
MSC 2010: 65F30; 65F50; 65N35; 65F10

A Appendix

A.1 Overview on Block Circulant Matrices

We recall that a one-level block circulant matrix A∈ℬ⁢𝒞⁢(L,m0) is defined by

(A.1)A=bcirc⁡{A0,A1,…,AL-1}=[A0AL-1⋯A2A1A1A0⋯⋮A2⋮⋮⋱A0⋮AL-1AL-2⋯A1A0]∈ℝL⁢m0×L⁢m0,

where Ak∈ℝm0×m0 for k=0,1,…,L-1 are matrices of general structure (see [14]). The equivalent Kronecker product representation is defined by the associated matrix polynomial,

(A.2)A=∑k=0L-1πk⊗Ak=:pA(π),

where π=πL∈ℝL×L is the periodic downward shift (cycling permutation) matrix,

(A.3)πL:=[00⋯0110⋯00⋮⋮⋱⋮⋮0⋯10000⋯10],

and ⊗ denotes the Kronecker product of matrices.

In the case m0=1, a matrix A∈ℬ⁢𝒞⁢(L,1) defines a circulant matrix generated by its first column vector a^=(a0,…,aL-1)T. The associated scalar polynomial then reads

pA⁢(z):=a0+a1⁢z+⋯+aL-1⁢zL-1,

so that (A.2) simplifies to

A=pA⁢(πL).

Let ω=ωL=exp⁡(-2⁢π⁢iL) and denote by

FL={fk⁢ℓ}∈ℝL×L with ⁢fk⁢ℓ=1L⁢ωL(k-1)⁢(ℓ-1),k,ℓ=1,…,L,

the unitary matrix of Fourier transform. Since the shift matrix πL is diagonalizable in the Fourier basis,

(A.4)πL=FL∗⁢DL⁢FL,DL=diag⁡{1,ω,…,ωL-1},

the same holds for any circulant matrix,

(A.5)A=pA⁢(πL)=FL∗⁢pA⁢(DL)⁢FL,

where

pA⁢(DL)=diag⁡{pA⁢(1),pA⁢(ω),…,pA⁢(ωL-1)}=diag⁡{FL⁢a}.

Conventionally, we denote by diag⁡{x} a diagonal matrix generated by a vector x. Let X be an L⁢m0×m0 matrix obtained by concatenation of m0×m0 matrices Xk, k=0,…,L-1,

X=conc⁡(X0,…,XL-1)=[X0,…,XL-1]T.

For example, the first block column in (A.1) has the form conc⁡(A0,…,AL-1). We denote by bdiag⁡{X} the L⁢m0×L⁢m0 block-diagonal matrix of block size L generated by m0×m0 blocks Xk.

It is known that similarly to the case of circulant matrices (A.5), a block circulant matrix in ℬ⁢𝒞⁢(L,m0) is unitary equivalent to the block diagonal one by means of Fourier transform via representation (A.2), see [14]. In the following, we describe the block-diagonal representation of a matrix A∈ℬ⁢𝒞⁢(L,m0) in the form that is convenient for generalization to the multi-level block circulant matrices as well as for the description of FFT-based implementation schemes. To that end, let us introduce the reshaping (folding) transform 𝒯L that maps an L⁢m0×m0 matrix X (i.e., the first block column in A) to the L×m0×m0 tensor B=𝒯L⁢X by plugging the ith m0×m0 block in X into a slice B⁢(i,:,:). The respective unfolding returns the initial matrix X=𝒯L′⁢B. We denote by A^∈ℝL⁢m0×m0 the first block column of a matrix A∈ℬ⁢𝒞⁢(L,m0), with a shorthand notation

A^=[A0,A1,…,AL-1]T,

so that the L×m0×m0 tensor 𝒯L⁢A^ represents slice-wise all generating m0×m0 matrix blocks.

Proposition A.1.

For A∈B⁢C⁢(L,m0) we have

A=(FL∗⊗Im0)⁢bdiag⁡{A¯0,A¯1,…,A¯L-1}⁢(FL⊗Im0),

where

A¯j=∑k=0L-1ωLj⁢k⁢Ak∈ℂm0×m0

can be recognized as the j-th m0×m0 matrix block in block column TL′⁢(FL⁢(TL⁢A^)), such that

[A¯0,A¯1,…,A¯L-1]T=𝒯L′⁢(FL⁢(𝒯L⁢A^)).

A set of eigenvalues λ of A is then given by

{λ∣A⁢x=λ⁢x,x∈ℂL⁢m0}=⋃j=0L-1{λ∣A¯j⁢u=λ⁢u,u∈ℂm0}.

The eigenvectors corresponding to the spectral sets

Σj={λj,m∣A¯j⁢uj,m=λj,m⁢uj,m,uj,m∈ℂm0,m=1,…,m0},j=0,1,…,L-1,

can be represented in the form

Uj,m=(FL∗⊗Im)⁢U¯j,m,where ⁢U¯j,m=E[j]⁢vec⁡[u0,m,u1,m,…,uL-1,m],

with E[j]=diag⁡{ej}⊗Im0∈RL⁢m0×L⁢m0, and ej∈RL being the jth Euclidean basis vector.

Proof.

We combine representations (A.2) and (A.4) to obtain

A=∑k=0L-1πk⊗Ak=∑k=0L-1(FL∗⁢Dk⁢FL)⊗Ak
=(FL∗⊗Im0)⁢(∑k=0L-1Dk⊗Ak)⁢(FL⊗Im0)
=(Fn∗⊗Im)⁢(∑k=0L-1bdiag⁡{Ak,ωLk⁢Ak,…,ωLk⁢(L-1)⁢Ak})⁢(FL⊗Im0)
=(FL∗⊗Im0)⁢bdiag⁡{∑k=0L-1Ak,∑k=0L-1ωLk⁢Ak,…,∑k=0L-1ωLk⁢(L-1)⁢Ak}⁢(FL⊗Im0)
=(FL∗⊗Im0)⁢bdiagm0×m0⁡{𝒯L′⁢(FL⁢(𝒯L⁢A^))}⁡(FL⊗Im0),

where the final step follows by the definition of FT matrix and by the construction of 𝒯L. The structure of eigenvalues and eigenfunctions then follows by straightforward calculations with block-diagonal matrices. This concludes the proof. ∎

The next statement describes the block-diagonal form for a class of symmetric BC matrices, ℬ⁢𝒞s⁢(L,m0). It is a simple corollary of [14] and Proposition A.1. In this case we have A0=A0T, and AkT=AL-k, k=1,…,L-1.

Corollary A.2.

Let A∈B⁢Cs⁢(L,m0) be symmetric. Then A is unitary similar to a Hermitian block-diagonal matrix, i.e., A is of the form

(A.6)A=(FL⊗Im0)⁢bdiag⁡(A~0,A~1,…,A~L-1)⁢(FL∗⊗Im0),

where Im0 is the m0×m0 identity matrix. The matrices A~j∈Cm0×m0, j=0,1,…,L-1, are defined for even n≥2 as

(A.7)A~j=A0+∑k=1L/2-1(ωLk⁢j⁢Ak+ω^Lk⁢j⁢AkT)+(-1)j⁢AL/2.

Corollary A.2 combined with Proposition A.1 describes a simplified structure of spectral data in the symmetric case. Notice that the above representation imposes the symmetry of each real-valued diagonal block A~j∈ℝm0×m0, j=0,1,…,L-1, in (A.6).

A.2 Multilevel Block Circulant/Toeplitz Matrices

We describe the extension of (one-level) block circulant matrices to multilevel structure. First, we recall the main notions of multilevel block circulant (MBC) matrices with the particular focus on the three-level case. Given a multi-index 𝐋=(L1,L2,L3), we denote |𝐋|=L1⁢L2⁢L3. A matrix class ℬ⁢𝒞⁢(d,𝐋,m0) (d=1,2,3) of d-level block circulant matrices can be introduced by the following recursion.

Definition A.3.

For d=1, define a class of one-level block circulant matrices by ℬ⁢𝒞⁢(1,𝐋,m)≡ℬ⁢𝒞⁢(L1,m) (see Appendix A.1), where 𝐋=(L1,1,1). For d=2, we say that a matrix A∈ℝ|𝐋|⁢m0×|𝐋|⁢m0 belongs to a class ℬ⁢𝒞⁢(d,𝐋,m0) if

A=bcirc⁡(A1,…,AL1) with ⁢Aj∈ℬ⁢𝒞⁢(d-1,𝐋[1],m0),j=1,…,L1,

where 𝐋[1]=(L2,L3)∈ℕd-1. A similar recursion applies to the case d=3.

Likewise to the case of one-level BC matrices, it can be seen that a matrix A∈ℬ⁢𝒞⁢(d,𝐋,m0), d=1,2,3, of size |𝐋|⁢m0×|𝐋|⁢m0 is completely defined (parametrized) by a dth order matrix-valued tensor 𝐀=[Ak1⁢…⁢kd] of size L1×⋯×Ld (kℓ=1,…,Lℓ, ℓ=1,…,d) with m0×m0 matrix entries Ak1⁢…⁢kd, obtained by folding the generating first column vector in A.

Recall that a symmetric block Toeplitz matrix A∈ℬ⁢𝒯s⁢(L,m0) is defined by

A=BToepls⁡{A0,A1,…,AL-1}=[A0A1T⋯AL-2TAL-1TA1A0⋯⋮AL-2T⋮⋮⋱A0⋮AL-1AL-2⋯A1A0]∈ℝL⁢m0×L⁢m0,

where Ak∈ℝm0×m0 for k=0,1,…,L-1 is a matrix of a general structure (see [14]).

A matrix class ℬ⁢𝒯s⁢(d,𝐋,m0) of symmetric d-level block Toeplitz matrices can be introduced by the following recursion, similarly to Definition A.3.

Definition A.4.

For d=1, ℬ⁢𝒯s⁢(1,𝐋,m0)≡ℬ⁢𝒯s⁢(L1,m0) is the class of one-level symmetric block circulant matrices with 𝐋=(L1,1,1). For d=2 we say that a matrix A∈ℝ|𝐋|⁢m×|𝐋|⁢m0 belongs to a class ℬ⁢𝒯s⁢(d,𝐋,m0) if

A=btoepls⁡(A1,…,AL1) with ⁢Aj∈ℬ⁢𝒯s⁢(d-1,𝐋[𝟏],m0),j=1,…,L1.

A similar recursion applies to the case d=3.

A.3 Rank-Structured Tensor Formats

We consider a tensor of order d, as a multidimensional array numbered by a d-tuple index set, 𝐀=[ai1,…,id]∈ℝn1×⋯×nd. A tensor is an element of a linear vector space equipped with the Euclidean scalar product. In particular, a tensor with equal sizes nℓ=n, ℓ=1,…,d, is called an n⊗d tensor. The required storage for entry-wise representation of tensors scales exponentially in the dimension, nd (the so-called “curse of dimensionality”). To get rid of exponential scaling in the dimension, one can apply the rank-structured separable representations of multidimensional tensors.

The rank-1 canonical tensor 𝐀=𝐮(1)⊗⋯⊗𝐮(d)∈ℝn1×⋯×nd with entries ai1,…,id=ai1(1)⁢⋯⁢aid(d) requires only d⁢n numbers to store it. A tensor in the R-term canonical format (CP tensors) is defined by the parametrization

𝐀=∑k=1Rck⁢𝐮k(1)⊗⋯⊗𝐮k(d),ck∈ℝ,

where uk(ℓ) are normalized vectors, and R is called the canonical rank of a tensor. The storage size is bounded by d⁢n⁢R≪nd.

Given the rank parameter 𝐫=(r1,…,rd), a tensor in the rank-𝐫 Tucker format is defined by the parametrization

𝐀=∑ν1=1r1…⁢∑νd=1rdβν1,…,νd⁢𝐯ν1(1)⊗⋯⊗𝐯νd(d),ℓ=1,…,d,

completely specified by a set of orthonormal vectors 𝐯νℓ(ℓ)∈ℝnℓ, and the Tucker core tensor 𝜷=[βν1,…,νd]. The storage demand is bounded by |𝐫|+(r1+⋯+rd)⁢n.

The remarkable approximating properties of the Tucker and canonical tensor decomposition applied to the wide class of function-related tensors were revealed in [39, 27, 42], promoting using tensor tools for the numerical treatment of the multidimensional PDEs. It was proved for some classes of function related tensors that the rank-𝐫 Tucker approximation provides the exponentially small error of the order of e-α⁢r with r=min⁡rℓ, where r=O⁢(log⁡n) (see [39]).

In the case of many spacial dimensions, the product type tensor formats provide the stable rank-structured approximation. The matrix-product states (MPS) decomposition has been for a long time used in quantum chemistry and quantum information theory, see the survey paper [57]. The particular case of MPS representation is called a tensor train (TT) format [51, 50]. The quantics-TT (QTT) tensor approximation method for functional n-vectors was introduced in [40] and shown to provide the logarithmic complexity, O⁢(d⁢log⁡n), on the wide class of generating functions. Furthermore, a combination of different tensor formats proved to be successful in the numerical solution of the multidimensional PDEs [41, 15].

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Received: 2017-1-30
Revised: 2017-3-25
Accepted: 2017-3-27
Published Online: 2017-5-31
Published in Print: 2017-7-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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Computational Methods in Applied Mathematics
Computational Methods in Applied Mathematics
Volume 17 Issue 3

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