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 $N$ 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 ($O(\log N)$) functional calls compared with the full grid size $N$. 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

ALADIN-α\alpha -- An open-source MATLAB toolbox for distributed non-convex optimization

This paper introduces an open-source software for distributed and decentralized non-convex optimization named ALADIN-$\alpha$. ALADIN-$\alpha$ is a MATLAB implementation of the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) algorithm, which is tailored towards rapid prototyping for non-convex distributed optimization. An improved version of the recently proposed bi-level variant of ALADIN is included enabling decentralized non-convex optimization. A collection of application examples from different applications fields including chemical engineering, robotics, and power systems underpins the application potential of ALADIN-$\alpha$

ALADIN-α—An open-source MATLAB toolbox for distributed non-convex optimization

This article introduces an open-source software for distributed and decentralized non-convex optimization named ALADIN-α. ALADIN-α is a MATLAB implementation of tailored variants of the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) algorithm. It is user interface is convenient for rapid prototyping of non-convex distributed optimization algorithms. An improved version of the recently proposed bi-level variant of ALADIN is included enabling decentralized non-convex optimization with reduced information exchange. A collection of examples from different applications fields including chemical engineering, robotics, and power systems underpins the potential of ALADIN-α

Fast iterative solution of the Bethe–Salpeter eigenvalue problem using low-rank and QTT tensor approximation

In this paper, we study and implement the structural iterative eigensolvers for the large-scale eigenvalue problem in the Bethe-Salpeter equation (BSE) based on the reduced basis approach via low-rank factorizations in generating matrices, introduced in the previous paper. The approach reduces numerical costs down to $\mathcal{O}(N_b^2)$ in the size of atomic orbitals basis set, $N_b$, instead of practically intractable $\mathcal{O}(N_b^6)$ complexity scaling for the direct diagonalization of the BSE matrix. As an alternative to rank approximation of the static screen interaction part of the BSE matrix, we propose to restrict it to a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate that the enhanced reduced-block approximation exhibits higher precision within the controlled numerical cost, providing as well a distinct two-sided error estimate for the BSE eigenvalues. It is shown that further reduction of the asymptotic computational cost is possible due to ALS-type iteration in block tensor train (TT) format applied to the quantized-TT (QTT) tensor representation of both long eigenvectors and rank-structured matrix blocks. The QTT-rank of these entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, $N_o$, hence the overall asymptotic complexity for solving the BSE problem can be estimated by $\mathcal{O}(\log(N_o) N_o^{2})$. We confirm numerically a considerable decrease in computational time for the presented iterative approach applied to various compact and chain-type molecules, while supporting sufficient accuracy.Comment: 23 pages, 11 figure