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.

本文提出了一种使用张量Krylov投影技术的通用框架来求解高阶Sylvester张量方程。在描述了广义Hessenberg过程的张量格式之后，我们将获得的不同过程与Galerkin正交条件或最小范数条件相结合，以得出Hess${}_{-}$BTF和CMRH${}_{-}$基于Hessenberg过程的张量格式的BTF方法。此外，我们还恢复了FOM${}_{-}$BTF和GMRES${}_{-}$BTF是基于Arnoldi过程的张量格式的已知方法。为了加快收敛或防止所获得的不同方法可能出现停滞，我们在建立张量Krylov子空间的基础时采用了基于加权内积而不是经典内积的加权策略。描述了数值实验，以便比较Hess提出的新方法${}_{-}$BTF和CMRH${}_{-}$使用已知方法BOM的BTF${}_{-}$BTF和GMRES${}_{-}$BTF并显示加权策略的效率。此外，我们针对所提出方法的未加权和加权形式采用了灵活的预处理框架，并且通过令人满意的数值结果验证了该灵活的版本。