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

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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 languageEnglish
Pages (from-to)149-172
Number of pages24
JournalRussian Journal of Numerical Analysis and Mathematical Modelling
Issue number2
Publication statusPublished - 5 Apr 2013

Bibliographical 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.

ASJC Scopus subject areas

  • Numerical Analysis
  • Modelling and Simulation


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