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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.

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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

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Date deposited: 17 Jun 2013 10:38
Last modified: 02 Dec 2019 20:49

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Contributors

Author: Sergey Dolgov
Author: Dmitry Savostyanov

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