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

By Sergey V. Dolgov. 2012

93 citations recorded by the source corpus.

A adapted tensor-structured GMRES method for the TT format is proposed and investigated. The Tensor Train (TT) approximation is a robust approach to high-dimensional problems. One class of problems is 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.

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Citations in the corpus

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, methodology): “For example, in the GMRES method [Dolgov, 2013], even if the solution vector and the right-hand side vector are well approximated by the TT format, the residuals and Krylov vectors involved in intermediate iterations usually have high TT ranks.”
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, unknown): “Following this paradigm, the solution of linear problems in tensor product formats was addressed in [36, 1, 6].”