On applying deflation and flexible preconditioning to the adaptive Simpler GMRES method for Sylvester tensor equations

Due to their multidimensional structure, the use of tensors when solving linear matrix equations has been one of the most explored topics over the last decade. In particular, the development of Krylov subspace methods for tensor equations is currently attracting a great deal of interest. In this work, we develop the adaptive simpler GMRES method based on tensor format (Ad-SGMRES${}_{-}$BTF) for solving Sylvester tensor equations, in order to address the numerical instability of the simpler version of GMRES based on tensor format (SGMRES${}_{-}$BTF). In addition, an efficient Krylov subspace method, formed by the integration of adaptive deflation and preconditioning and called FAd-SGMRES-D${}_{-}$BTF$(m,k)$ -in which $m$ is the restart parameter and $k$ is the number of harmonic Ritz pairs-, is proposed. Note that the idea behind the FAd-SGMRES-D${}_{-}$BTF$(m,k)$ method is to reuse certain key information, which can be obtained from the current tensor Krylov subspace in the next cycle of the FAd-SGMRES-D${}_{-}$BTF$\left(m\right)$ method. It is also important to note that FAd-SGMRES-D${}_{-}$BTF$(m,k)$ represents the first attempt to synchronously combine flexible preconditioning and deflation techniques, with the aim of improving the convergence rate and reducing the computational cost of Ad-SGMRES${}_{-}$BTF$\left(m\right)$. Finally, the computational cost of each cycle of the proposed methods when applied to a three-dimensional Sylvester tensor equation is estimated. Moreover, numerical experiments on synthetic and real-world applications demonstrate the potential of Ad-SGMRES${}_{-}$BTF and FAd-SGMRES-D${}_{-}$BTF$\left(m\right)$ to solve Sylvester tensor equations.