Solution of linear systems and matrix inversion in the TT-format

I. V. Oseledets, S. V. Dolgov

Research output: Contribution to journalArticlepeer-review

96 Citations (Scopus)

Abstract

Tensors arise naturally in high-dimensional pro blems in chemistry, financial mathematics, and many other areas. The numerical treatment of such problems is difficult due to the curse of dimensionality: the number of unknowns and the computational complexity grow exponentially with the dimension of the problem. To break the curse of dimensionality, low-parametric representations, or formats, have to be used. In this paper we make use of the TT-format (tensor-train format) which is one of the most effective stable representations of high-dimensional tensors. Basic linear algebra operations in the TT-format are now well developed. Our goal is to provide a "black-box" type of solver for linear systems where both the matrix and the right-hand side are in the TT-format. An efficient DMRG (density matrix renormalization group) method is proposed, and several tricks are employed to make it work. The numerical experiments confirm the effectiveness of our approach.

Original languageEnglish
Pages (from-to)A2718-A2739
JournalSIAM Journal on Scientific Computing
Volume34
Issue number5
DOIs
Publication statusPublished - 2012
Externally publishedYes

Keywords

  • Density matrix renormalization group
  • High-dimensional problems
  • Solution of linear systems
  • TT-format

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