Some iterative approaches for Sylvester tensor equations, Part I: A tensor format of truncated Loose Simpler GMRES

Abstract

In recent years, Krylov subspace methods based on the tensor format have demonstrated their superiority over classical ones for handling various kinds of tensor equations. In this paper, according to the efficient computational cost of the classical Simpler GMRES method (SGMRES), we adopt the tensor format of this method, which is called SGMRES−BTF, for solving the Sylvester tensor equations. This paper is divided into two parts describing the tensor format of two types of acceleration approaches to tackle some serious problems of the restarted methods GMRES−BTF and SGMRES−BTF, such as the “stalling” and “alternating phenomenon”. In the first part of this two-part work, the truncated Loose Simpler GMRES based on the tensor format (LSGMRES−BTF) is introduced, which is an acceleration approach that aims to construct an augmented tensor Krylov subspace by means of approximation error tensors. Moreover, a detailed study is carried out on the connection between the values of sequential and skip angles and the convergence behavior of both SGMRES−BTF and truncated LSGMRES−BTF. In the second part of this paper, an acceleration approach based on the idea of inner-outer iteration in the truncated version of the generalized conjugate residual with inner orthogonalization (GCRO) method is developed. In this method, SGMRES−BTF is applied in the inner iteration, and the generalized conjugate residual based on the tensor format (GCR−BTF) method is used in the outer iteration. Numerical experiments show that SGMRES−BTF achieves an appropriate performance compared with GMRES−BTF. In addition, the numerical results reveal the high potential of the presented accelerating strategies to deal with Sylvester tensor equations.