TT-GMRES: Solution to a linear system in the structured tensor format

S. V. Dolgov

doi:10.1515/rnam-2013-0009

TT-GMRES: Solution to a linear system in the structured tensor format

S. V. Dolgov

Research output: Contribution to journal › Article › peer-review

23 Citations (SciVal)

Abstract

An adapted tensor-structured GMRES method for the TT format is proposed and investigated. The Tensor Train (TT) approximation is a robust approach to highdimensional problems. One class of such problems involves the solution of a linear system. In this work we study the convergence of the GMRES method in the presence of tensor approximations and provide relaxation techniques to improve its performance. Several numerical examples are presented. The method is also compared with a projection TT linear solver based on the ALS and DMRG methods. On a particular SPDE (high-dimensional parametric) problem these methods manifest comparable performance, with a good preconditioner the TT-GMRES overcomes the ALS solver.

Original language	English
Pages (from-to)	149-172
Number of pages	24
Journal	Russian Journal of Numerical Analysis and Mathematical Modelling
Volume	28
Issue number	2
DOIs	https://doi.org/10.1515/rnam-2013-0009
Publication status	Published - 5 Apr 2013

ASJC Scopus subject areas

Numerical Analysis
Modelling and Simulation

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10.1515/rnam-2013-0009

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abstract = "An adapted tensor-structured GMRES method for the TT format is proposed and investigated. The Tensor Train (TT) approximation is a robust approach to highdimensional problems. One class of such problems involves the solution of a linear system. In this work we study the convergence of the GMRES method in the presence of tensor approximations and provide relaxation techniques to improve its performance. Several numerical examples are presented. The method is also compared with a projection TT linear solver based on the ALS and DMRG methods. On a particular SPDE (high-dimensional parametric) problem these methods manifest comparable performance, with a good preconditioner the TT-GMRES overcomes the ALS solver.",

author = "Dolgov, {S. V.}",

note = "Funding Information: Russia This work was supported in part by RFBR grants 09-01-12058, 10-01-00757, 11-01-00549, RFBR/DFG grant 09-01-91332, Russian Federation Gov. contracts No. P1178, P1112 and P940, 14.740.11.0345 and Promotionstipendium of Max-Planck Gesellschaft. Part of this work was done during the stay of the author at Max-Plank Institute for Mathematics in Sciences, Leipzig, Germany.",

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AB - An adapted tensor-structured GMRES method for the TT format is proposed and investigated. The Tensor Train (TT) approximation is a robust approach to highdimensional problems. One class of such problems involves the solution of a linear system. In this work we study the convergence of the GMRES method in the presence of tensor approximations and provide relaxation techniques to improve its performance. Several numerical examples are presented. The method is also compared with a projection TT linear solver based on the ALS and DMRG methods. On a particular SPDE (high-dimensional parametric) problem these methods manifest comparable performance, with a good preconditioner the TT-GMRES overcomes the ALS solver.

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ER -

TT-GMRES: Solution to a linear system in the structured tensor format

Abstract

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