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 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 lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic 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.1 Overview on Block Circulant Matrices
We recall that a one-level block circulant matrix is defined by
where for are matrices of general structure (see ). The equivalent Kronecker product representation is defined by the associated matrix polynomial,
where is the periodic downward shift (cycling permutation) matrix,
and denotes the Kronecker product of matrices.
In the case , a matrix defines a circulant matrix generated by its first column vector . The associated scalar polynomial then reads
so that (A.2) simplifies to
Let and denote by
the unitary matrix of Fourier transform. Since the shift matrix is diagonalizable in the Fourier basis,
the same holds for any circulant matrix,
Conventionally, we denote by a diagonal matrix generated by a vector x. Let X be an matrix obtained by concatenation of matrices , ,
For example, the first block column in (A.1) has the form . We denote by the block-diagonal matrix of block size L generated by blocks .
It is known that similarly to the case of circulant matrices (A.5), a block circulant matrix in is unitary equivalent to the block diagonal one by means of Fourier transform via representation (A.2), see . In the following, we describe the block-diagonal representation of a matrix 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 that maps an matrix X (i.e., the first block column in A) to the tensor by plugging the ith block in X into a slice . The respective unfolding returns the initial matrix . We denote by the first block column of a matrix , with a shorthand notation
so that the tensor represents slice-wise all generating matrix blocks.
For we have
can be recognized as the j-th matrix block in block column , such that
A set of eigenvalues λ of A is then given by
The eigenvectors corresponding to the spectral sets
can be represented in the form
with , and being the th Euclidean basis vector.
where the final step follows by the definition of FT matrix and by the construction of . The structure of eigenvalues and eigenfunctions then follows by straightforward calculations with block-diagonal matrices. This concludes the proof. ∎
Let be symmetric. Then is unitary similar to a Hermitian block-diagonal matrix, i.e., is of the form
where is the identity matrix. The matrices , , are defined for even as
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 , , 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 , we denote . A matrix class () of -level block circulant matrices can be introduced by the following recursion.
For , define a class of one-level block circulant matrices by (see Appendix A.1), where . For , we say that a matrix belongs to a class if
where . A similar recursion applies to the case .
Likewise to the case of one-level BC matrices, it can be seen that a matrix , , of size is completely defined (parametrized) by a th order matrix-valued tensor of size (, ) with matrix entries , obtained by folding the generating first column vector in .
Recall that a symmetric block Toeplitz matrix is defined by
where for is a matrix of a general structure (see ).
A matrix class of symmetric -level block Toeplitz matrices can be introduced by the following recursion, similarly to Definition A.3.
For , is the class of one-level symmetric block circulant matrices with . For we say that a matrix belongs to a class if
A similar recursion applies to the case .
A.3 Rank-Structured Tensor Formats
We consider a tensor of order , as a multidimensional array numbered by a -tuple index set, . A tensor is an element of a linear vector space equipped with the Euclidean scalar product. In particular, a tensor with equal sizes , , is called an tensor. The required storage for entry-wise representation of tensors scales exponentially in the dimension, (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- canonical tensor with entries requires only numbers to store it. A tensor in the -term canonical format (CP tensors) is defined by the parametrization
where are normalized vectors, and is called the canonical rank of a tensor. The storage size is bounded by .
Given the rank parameter , a tensor in the rank- Tucker format is defined by the parametrization
completely specified by a set of orthonormal vectors , and the Tucker core tensor . The storage demand is bounded by .
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- Tucker approximation provides the exponentially small error of the order of with , where (see ).
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 . The particular case of MPS representation is called a tensor train (TT) format [51, 50]. The quantics-TT (QTT) tensor approximation method for functional -vectors was introduced in  and shown to provide the logarithmic complexity, , 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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