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In the second part of this two-part work, another accelerator method based on the tensor format is established for solving the Sylvester tensor equations. This acceleration approach, which is called SGCRO−BTF, is based on the idea of inner-outer iteration used in the generalized conjugate residual with inner orthogonalization (GCRO) method. In SGCRO−BTF, the Simpler GMRES method based on the tensor format (SGMRES−BTF) is applied to the inner iteration, and the generalized conjugate residual based on the tensor format (GCR−BTF) method is used in the outer iteration. Furthermore, SGCRO−BTF seeks an approximate solution over a tensor subspace spanned by the approximation error tensors produced during the previous outer iterations of SGCRO−BTF and a tensor Krylov subspace constructed by the inner iteration. In order to reduce the computational storage in the outer iteration, the truncated version of SGCRO−BTF is presented, in which only some of the last approximation error tensors are kept and will be then added to the tensor Krylov subspace in order to obtain a new search subspace. Finally, the proposed methods are tested on a set of experiments and compared to some conventional and state-of-the-art Krylov subspace methods based on the tensor format. Experimental results indicate that the truncated version of SGCRO−BTF is particularly effective for solving the Sylvester tensor equations. |
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