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

Sergey Dolgov and Ivan V. Oseledets


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.

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

Related publications

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