A tensor format for the generalized Hessenberg method for solving Sylvester tensor equations

In this paper, a general framework using tensor Krylov projection techniques is proposed for solving high order Sylvester tensor equations. After describing the tensor format of the generalized Hessenberg process, we combine the obtained different processes with a Galerkin orthogonality condition or with a minimal norm condition in order to derive the Hess_BTF and CMRH_BTF methods which are based on the tensor format of the Hessenberg process. In addition, we also recover the FOM_BTF and GMRES_BTF which are known methods based on the tensor format of the Arnoldi process. To accelerate the convergence or prevent a possible stagnation of the different obtained methods, we incorporate a weighting strategy based on the use of a weighted inner product instead of the classical one when building a basis for the tensor Krylov subspace. Numerical experiments are described in order to compare the new proposed methods that are Hess_BTF and CMRH_BTF with the known methods FOM_BTF and GMRES_BTF and to show the efficiency of the weighting strategy. Moreover, we utilize a flexible preconditioning framework for the unweighted and weighted forms of the proposed methods, and the flexible version is validated by satisfactory numerical results. (C) 2020 Elsevier B.V. All rights reserved.