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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
19/2011

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

Sergey Dolgov and Ivan V. Oseledets

Abstract

Tensors arise naturally in high-dimensional problems in chemistry, financial mathematics and many others. The numerical treatment of such kind of problems is difficult due to the curse of dimensionality: the number of unknowns and computational complexity grows 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 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 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.

Received:
Apr 27, 2011
Published:
Apr 28, 2011
MSC Codes:
65F30, 65F50, 65N35, 65F10
Keywords:
TT-format, solution of linear systems, high-dimensional problems, DMRG

Related publications

inJournal
2012 Repository Open Access
Ivan V. Oseledets and Sergey Dolgov

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

In: SIAM journal on scientific computing, 34 (2012) 5, A2718-A2739