11 research outputs found
Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach
This paper introduces and analyses the new grid-based tensor approach to
approximate solution 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 up to
several hundred confirm the theoretical complexity bounds for the
block-structured eigenvalue solvers in the limit of large .Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with
arXiv:1408.383
Computing the density of states for optical spectra by low-rank and QTT tensor approximation
In this paper, we introduce a new interpolation scheme to approximate the
density of states (DOS) for a class of rank-structured matrices with
application to the Tamm-Dancoff approximation (TDA) of the Bethe-Salpeter
equation (BSE). The presented approach for approximating the DOS is based on
two main techniques. First, we propose an economical method for calculating the
traces of parametric matrix resolvents at interpolation points by taking
advantage of the block-diagonal plus low-rank matrix structure described in [6,
3] for the BSE/TDA problem. Second, we show that a regularized or smoothed DOS
discretized on a fine grid of size can be accurately represented by a low
rank quantized tensor train (QTT) tensor that can be determined through a least
squares fitting procedure. The latter provides good approximation properties
for strictly oscillating DOS functions with multiple gaps, and requires
asymptotically much fewer () functional calls compared with the full
grid size . This approach allows us to overcome the computational
difficulties of the traditional schemes by avoiding both the need of stochastic
sampling and interpolation by problem independent functions like polynomials
etc. Numerical tests indicate that the QTT approach yields accurate recovery of
DOS associated with problems that contain relatively large spectral gaps. The
QTT tensor rank only weakly depends on the size of a molecular system which
paves the way for treating large-scale spectral problems.Comment: 26 pages, 25 figure