87 research outputs found
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
Fast iterative solution of the Bethe–Salpeter eigenvalue problem using low-rank and QTT tensor approximation
In this paper, we propose and study two approaches to approximate the solution of the Bethe–Salpeter equation (BSE) by using structured iterative eigenvalue solvers. Both approaches are based on the reduced basis method and low-rank factorizations of the generating matrices. We also propose to represent the static screen interaction part in the BSE matrix by a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate by various numerical tests that the combination of the diagonal plus low-rank plus reduced-block approximation exhibits higher precision with low numerical cost, providing as well a distinct two-sided error estimate for the smallest eigenvalues of the Bethe–Salpeter operator. The complexity is reduced to O(Nb 2) in the size of the atomic orbitals basis set, Nb, instead of the practically intractable O(Nb 6) scaling for the direct diagonalization. In the second approach, we apply the quantized-TT (QTT) tensor representation to both, the long eigenvectors and the column vectors in the rank-structured BSE matrix blocks, and combine this with the ALS-type iteration in block QTT format. The QTT-rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, Nob, hence the overall asymptotic complexity for solving the BSE problem by the QTT approximation is estimated by O(log(No)No 2). We confirm numerically a considerable decrease in computational time for the presented iterative approaches applied to various compact and chain-type molecules, while supporting sufficient accuracy.</p
ALADIN- -- 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-. ALADIN- 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-
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 in the size of atomic orbitals basis set,
, instead of practically intractable 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, , hence the
overall asymptotic complexity for solving the BSE problem can be estimated by
. 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
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