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

Venera Khoromskaia and Boris N. Khoromskij

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.

A Appendix

A.1 Overview on Block Circulant Matrices

We recall that a one-level block circulant matrix A𝒞(L,m0) is defined by

(A.1)A=bcirc{A0,A1,,AL-1}=[A0AL-1A2A1A1A0A2A0AL-1AL-2A1A0]Lm0×Lm0,

where Akm0×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-1πkAk=:pA(π),

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

(A.3)πL:=[0001100001000010],

and denotes the Kronecker product of matrices.

In the case m0=1, a matrix A𝒞(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+a1z++aL-1zL-1,

so that (A.2) simplifies to

A=pA(πL).

Let ω=ωL=exp(-2πiL) and denote by

FL={fk}L×Lwith fk=1LωL(k-1)(-1),k,=1,,L,

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

(A.4)πL=FLDLFL,DL=diag{1,ω,,ωL-1},

the same holds for any circulant matrix,

(A.5)A=pA(πL)=FLpA(DL)FL,

where

pA(DL)=diag{pA(1),pA(ω),,pA(ωL-1)}=diag{FLa}.

Conventionally, we denote by diag{x} a diagonal matrix generated by a vector x. Let X be an Lm0×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 Lm0×Lm0 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 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 A𝒞(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 Lm0×m0 matrix X (i.e., the first block column in A) to the L×m0×m0 tensor B=𝒯LX by plugging the ith m0×m0 block in X into a slice B(i,:,:). The respective unfolding returns the initial matrix X=𝒯LB. We denote by A^Lm0×m0 the first block column of a matrix A𝒞(L,m0), with a shorthand notation

A^=[A0,A1,,AL-1]T,

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

Proposition A.1.

For ABC(L,m0) we have

A=(FLIm0)bdiag{A¯0,A¯1,,A¯L-1}(FLIm0),

where

A¯j=k=0L-1ωLjkAkm0×m0

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

[A¯0,A¯1,,A¯L-1]T=𝒯L(FL(𝒯LA^)).

A set of eigenvalues λ of A is then given by

{λAx=λx,xLm0}=j=0L-1{λA¯ju=λu,um0}.

The eigenvectors corresponding to the spectral sets

Σj={λj,mA¯juj,m=λj,muj,m,uj,mm0,m=1,,m0},j=0,1,,L-1,

can be represented in the form

Uj,m=(FLIm)U¯j,m,where U¯j,m=E[j]vec[u0,m,u1,m,,uL-1,m],

with E[j]=diag{ej}Im0RLm0×Lm0, and ejRL being the jth Euclidean basis vector.

Proof.

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

A=k=0L-1πkAk=k=0L-1(FLDkFL)Ak
=(FLIm0)(k=0L-1DkAk)(FLIm0)
=(FnIm)(k=0L-1bdiag{Ak,ωLkAk,,ωLk(L-1)Ak})(FLIm0)
=(FLIm0)bdiag{k=0L-1Ak,k=0L-1ωLkAk,,k=0L-1ωLk(L-1)Ak}(FLIm0)
=(FLIm0)bdiagm0×m0{𝒯L(FL(𝒯LA^))}(FLIm0),

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, 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 ABCs(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=(FLIm0)bdiag(A~0,A~1,,A~L-1)(FLIm0),

where Im0 is the m0×m0 identity matrix. The matrices A~jCm0×m0, j=0,1,,L-1, are defined for even n2 as

(A.7)A~j=A0+k=1L/2-1(ωLkjAk+ω^LkjAkT)+(-1)jAL/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~jm0×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|=L1L2L3. A matrix class 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 𝒞(1,𝐋,m)𝒞(L1,m) (see Appendix A.1), where 𝐋=(L1,1,1). For d=2, we say that a matrix A|𝐋|m0×|𝐋|m0 belongs to a class 𝒞(d,𝐋,m0) if

A=bcirc(A1,,AL1)with Aj𝒞(d-1,𝐋[1],m0),j=1,,L1,

where 𝐋[1]=(L2,L3)d-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 A𝒞(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=[Ak1kd] of size L1××Ld (k=1,,L, =1,,d) with m0×m0 matrix entries Ak1kd, obtained by folding the generating first column vector in A.

Recall that a symmetric block Toeplitz matrix A𝒯s(L,m0) is defined by

A=BToepls{A0,A1,,AL-1}=[A0A1TAL-2TAL-1TA1A0AL-2TA0AL-1AL-2A1A0]Lm0×Lm0,

where Akm0×m0 for k=0,1,,L-1 is a matrix of a general structure (see [14]).

A matrix class 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, 𝒯s(1,𝐋,m0)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|𝐋|m×|𝐋|m0 belongs to a class 𝒯s(d,𝐋,m0) if

A=btoepls(A1,,AL1)with Aj𝒯s(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]n1××nd. A tensor is an element of a linear vector space equipped with the Euclidean scalar product. In particular, a tensor with equal sizes n=n, =1,,d, is called an nd 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)n1××nd with entries ai1,,id=ai1(1)aid(d) requires only dn numbers to store it. A tensor in the R-term canonical format (CP tensors) is defined by the parametrization

𝐀=k=1Rck𝐮k(1)𝐮k(d),ck,

where uk() are normalized vectors, and R is called the canonical rank of a tensor. The storage size is bounded by dnRnd.

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

𝐀=ν1=1r1νd=1rdβν1,,νd𝐯ν1(1)𝐯νd(d),=1,,d,

completely specified by a set of orthonormal vectors 𝐯ν()n, 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-αr with r=minr, where r=O(logn) (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(dlogn), 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