Computing the density of states for optical spectra of molecules by low-rank and QTT tensor approximation

Abstract

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. This allows us to overcome the computational difficulties of the traditional schemes since we avoid the construction of the matrix inverse and hence the need of stochastic sampling. Second, we show that a regularized or smoothed DOS discretized on a fine grid of size N can be accurately represented in a low rank quantized tensor train (QTT) format that can be determined through a least squares fitting procedure. The QTT tensor provides good approximation properties for strictly oscillating DOS functions with multiple gaps, in contrast to interpolation by problem independent functions like polynomials, trigonometric functions, etc. Moreover, the QTT approximant requires asymptotically much fewer (e.g., O(log⁡N)) functional calls compared with the full grid size N. 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 that paves the way for treatment of large-scale spectral problems.