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

摘要

An 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 such problems involves the 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.