Krylov subspace projection method for Sylvester tensor equation with low rank right-hand side
The Sylvester tensor equation (STE) is a popular problem in tensor-based computation. It can arise for example in discretizing high-dimensional linear partial differential equations using finite difference or spectral methods. STE is equivalent to a system of linear equations but solving this linear system can be a real challenge. This nicely written paper deals with approximate solutions to the STE with low rank right-hand sides obtained by Arnoldi-based methods. At first, some basic definitions, tensor notations, and common operations related to tensors adopted in this paper are introduced. Theoretical results for obtaining an approximate solution using the Arnoldi algorithm for the linear system where the right-hand side tensor is of rank one are presented. Then, using the fact that the tensor of order \(N\) can be decomposed by the CP decomposition, the authors propose two new approaches to extract approximate solutions to STE with low rank right-hand sides: block and global Krylov methods, namely Arnoldi-based algorithms. The first approach is based on the use of the block Arnoldi algorithm for the coefficient matrices in STE, which leads to the reduced Sylvester tensor. The second approach is based on the use of the global Arnoldi algorithm in order to obtain the low-dimensional Sylvester tensor. The algorithms are summarized in the paper. Besides, expressions of the corresponding residual tensors and their norms for each approach are given. Three numerical examples showing the effectiveness of the proposed approaches are presented.
scientific article; zbMATH DE number 7221106
scientific article