A hybrid preconditioner of banded matrix approximation and alternating direction implicit iteration for symmetric Sinc-Galerkin linear systems
A hybrid preconditioner of banded matrix approximation and alternating direction implicit iteration for symmetric Sinc-Galerkin linear systems
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DOI：
10.1016/s0024-3795(02)00502-5
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发表时间：
2003-06
影响因子：
1.1
通讯作者：
M. Ng;Z. Bai
中科院分区：
文献类型：
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作者：
M. Ng;Z. Bai
将对称Sinc-Galerkin方法应用于可稀疏的二阶自伴椭圆边值问题，得到一个线性方程组[公式：见正文]，其中λ是Kronecker乘积符号，λ x和λ y是Toeplitz加对角矩阵，Dx和Dy是对角矩阵。本文的主要贡献是提出并分析了两步预处理策略的基础上带状矩阵近似（BMA）和交替方向隐式（ADI）迭代这些Sinc-Galerkin系统。特别地，我们证明了两步预条件子是对称正定的，并且预条件矩阵的条件数由所涉及的ADI迭代的收敛因子有界。数值算例表明，该预条件子是求解上述对称Sinc-Galerkin线性方程组的共轭梯度法的有效预条件。
The symmetric Sinc–Galerkin method applied to a sparable second-order self-adjoint elliptic boundary value problem gives rise to a system of linear equations [Formula: see text] where⊗ is the Kronecker product symbol, Ψxand Ψyare Toeplitz-plus-diagonal matrices, and Dxand Dyare diagonal matrices. The main contribution of this paper is to present and analyze a two-step preconditioning strategy based on the banded matrix approximation (BMA) and the alternating direction implicit (ADI) iteration for these Sinc–Galerkin systems. In particular, we show that the two-step preconditioner is symmetric positive definite, and the condition number of the preconditioned matrix is bounded by the convergence factor of the involved ADI iteration. Numerical examples show that the new preconditioner is practical and efficient to precondition the conjugate gradient method for solving the above symmetric Sinc–Galerkin linear system.