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Register a new userOn circulant and skew-circulant splitting algorithms for (continuous) Sylvester equations
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Added by
Yulin Zhang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We present a circulant and skew-circulant splitting (CSCS) iterative method for solving large sparse continuous Sylvester equations $AX + XB = C$, where the coefficient matrices $A$ and $B$ are Toeplitz matrices. A theoretical study shows that if the circulant and skew-circulant splitting factors of $A$ and $B$ are positive semi-definite and at least one is positive definite (not necessarily Hermitian), then the CSCS method converges to the unique solution of the Sylvester equation. In addition, we obtain an upper bound for the convergence factor of the CSCS iteration. This convergence factor depends only on the eigenvalues of the circulant and skew-circulant splitting matrices. A computational comparison with alternative methods reveals the efficiency and reliability of the proposed method.
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Circulant preconditioners for functions of matrices have been recently of interest. In particular, several authors proposed the use of the optimal circulant preconditioners as well as the superoptimal circulant preconditioners in this context and numerically illustrated that such preconditioners are effective for certain functions of Toeplitz matrices. Motivated by their results, we propose in this work the absolute value superoptimal circulant preconditioners and provide several theorems that analytically show the effectiveness of such circulant preconditioners for systems defined by functions of Toeplitz matrices. Namely, we show that the eigenvalues of the preconditioned matrices are clustered around $pm 1$ and rapid convergence of Krylov subspace methods can therefore be expected. Moreover, we show that our results can be extended to functions of block Toeplitz matrices with Toeplitz blocks provided that the optimal block circulant matrices with circulant blocks are used as preconditioners. Numerical examples are given to support our theoretical results.
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This paper introduces and analyzes a preconditioned modified of the Hermitian and skew-Hermitian splitting (PMHSS). The large sparse continuous Sylvester equations are solved by PMHSS iterative algorithm based on nonHermitian, complex, positive definite/semidefinite, and symmetric matrices. We prove that the PMHSS is converged under suitable conditions. In addition, we propose an accelerated PMHSS method consisting of two preconditioned matrices and two iteration parameters {alpha}, b{eta}. Theoretical analysis showed that the convergence speed of the accelerated PMHSS is faster compared to the PMHSS. Also, the robustness and efficiency of the proposed two iterative algorithms were demonstrated in numerical experiments.
In this paper, we study temporal splitting algorithms for multiscale problems. The exact fine-grid spatial problems typically require some reduction in degrees of freedom. Multiscale algorithms are designed to represent the fine-scale details on a coarse grid and, thus, reduce the problems size. When solving time-dependent problems, one can take advantage of the multiscale decomposition of the solution and perform temporal splitting by solving smaller-dimensional problems, which is studied in the paper. In the proposed approach, we consider the temporal splitting based on various low dimensional spatial approximations. Because a multiscale spatial splitting gives a good decomposition of the solution space, one can achieve an efficient implicit-explicit temporal discretization. We present a recently developed theoretical result in our earlier work and adopt it in this paper for multiscale problems. Numerical results are presented to demonstrate the efficiency of the proposed splitting algorithm.
A Schur ring (S-ring) over a group $G$ is called separable if every of its similaritities is induced by isomorphism. We establish a criterion for an S-ring to be separable in the case when the group $G$ is cyclic. Using this criterion, we prove that any S-ring over a cyclic $p$-group is separable and that the class of separable circulant S-rings is closed with respect to duality.
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