SIAM Journal on Scientific Computing· 2012Q1
Solution of Linear Systems and Matrix Inversion in the TT-Format
- 195citations
- Q1SCImago
- 2012year
Short summary
A novel DMRG-based solver efficiently handles linear systems where both the matrix and right-hand side are represented in the tensor-train (TT) format, overcoming the curse of dimensionality.
AI-generated from the title and abstract; the full text is not read.
Key points
- Introduces a new DMRG-based solver for linear systems in TT-format.
- Handles both the matrix and right-hand side vector in TT-format.
- Addresses the 'curse of dimensionality' in high-dimensional problems.
- Numerical experiments validate the solver's effectiveness.
AI-generated from the title and abstract; the full text is not read.
Abstract
Tensors arise naturally in high-dimensional problems 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.
The authors' abstract, as published at the source. SIAM Journal on Scientific Computing, 2012 · DOI ↗
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Field: Computational Mathematics
Computational MathematicsMathematics