The University of Southampton
University of Southampton Institutional Repository

Alternating minimal energy methods for linear systems in higher dimensions

Alternating minimal energy methods for linear systems in higher dimensions
Alternating minimal energy methods for linear systems in higher dimensions
We propose algorithms for the solution of high--dimensional symmetrical positive definite (SPD) linear systems with the matrix and the right hand side given and the solution sought in a low-rank format. Similarly to density matrix renormalization group (DMRG) algorithms, our methods optimize the components of the tensor product format subsequently. To improve the convergence, we expand the search space by an inexact gradient direction. We prove the geometrical convergence and estimate the convergence rate of the proposed methods utilizing the analysis of the steepest descent algorithm. The complexity of the presented algorithms is linear in the mode size and dimension, and the demonstrated convergence is comparable to or even better than the one of the DMRG algorithm. In the numerical experiment we show that the proposed methods are also efficient for non-SPD systems, for example those arising from the chemical master equation (CME) describing the gene regulatory model at the mesoscopic scale.
high-dimensional problems, tensor train format, alternating linear scheme, density matrix renormalization group, steepest descent, Poisson equation, chemical master equation
1064-8275
1-24
Dolgov, Sergey
a6b5facb-530c-4324-bc62-73f81ea00e23
Savostyanov, Dmitry
49d88c5f-b159-4dff-af88-5b9a5ff18322
Dolgov, Sergey
a6b5facb-530c-4324-bc62-73f81ea00e23
Savostyanov, Dmitry
49d88c5f-b159-4dff-af88-5b9a5ff18322

Dolgov, Sergey and Savostyanov, Dmitry (2014) Alternating minimal energy methods for linear systems in higher dimensions. SIAM Journal on Scientific Computing, 36 (5), 1-24. (doi:10.1137/140953289).

Record type: Article

Abstract

We propose algorithms for the solution of high--dimensional symmetrical positive definite (SPD) linear systems with the matrix and the right hand side given and the solution sought in a low-rank format. Similarly to density matrix renormalization group (DMRG) algorithms, our methods optimize the components of the tensor product format subsequently. To improve the convergence, we expand the search space by an inexact gradient direction. We prove the geometrical convergence and estimate the convergence rate of the proposed methods utilizing the analysis of the steepest descent algorithm. The complexity of the presented algorithms is linear in the mode size and dimension, and the demonstrated convergence is comparable to or even better than the one of the DMRG algorithm. In the numerical experiment we show that the proposed methods are also efficient for non-SPD systems, for example those arising from the chemical master equation (CME) describing the gene regulatory model at the mesoscopic scale.

Text
2014-ds-amen.pdf - Other
Download (908kB)

More information

Submitted date: 17 January 2014
Published date: 25 September 2014
Keywords: high-dimensional problems, tensor train format, alternating linear scheme, density matrix renormalization group, steepest descent, Poisson equation, chemical master equation
Organisations: Chemistry

Identifiers

Local EPrints ID: 353745
URI: http://eprints.soton.ac.uk/id/eprint/353745
ISSN: 1064-8275
PURE UUID: 748014ce-253c-40f2-95c8-869cb89495ef

Catalogue record

Date deposited: 17 Jun 2013 10:38
Last modified: 14 Mar 2024 14:09

Export record

Altmetrics

Contributors

Author: Sergey Dolgov
Author: Dmitry Savostyanov

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×