Preconditioning techniques for diagonal-times-toeplitz matrices in fractional diffusion equations
Preconditioning techniques for diagonal-times-toeplitz matrices in fractional diffusion equations
复制标题
分数扩散方程中对角次托普利茨矩阵的预处理技术
DOI：
10.1137/130931795
复制
发表时间：
2014
影响因子：
3.1
通讯作者：
Sun Hai-Wei
中科院分区：
文献类型：
--
作者：
Pan Jianyu;Ke Rihuan;Ng Michael K.;Sun Hai-Wei
分数阶扩散方程采用隐式有限差分格式离散，差分格式采用无条件稳定的位移Gru nwald公式.离散化线性系统的系数矩阵等于一个比例单位矩阵和两个对角时间Toeplitz矩阵之和。标准循环预条件子可能不适用于此类Toeplitz类线性系统。本文的主要目的是提出和发展近似逆预条件，这样的Toeplitz矩阵。构造了一个近似逆预条件子，将加权Toeplitz矩阵的逆矩阵用循环矩阵近似，然后逐行联合收割机合并.由于Toeplitz结构，离散化系数矩阵和预处理器可以非常有效地实现通过使用快速傅立叶变换。从理论上讲，我们表明，所得到的预处理矩阵的频谱是围绕一个集群。因此，Krylov子空间方法与建议的预条件收敛非常快。数值算例验证了该预条件子的有效性，并表明其性能优于其他测试预条件子。
The fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal-times-Toeplitz matrices. Standard circulant preconditioners may not work for such Toeplitz-like linear systems. The main aim of this paper is to propose and develop approximate inverse preconditioners for such Toeplitz-like matrices. An approximate inverse preconditioner is constructed to approximate the inverses of weighted Toeplitz matrices by circulant matrices, and then combine them together row-by-row. Because of Toeplitz structure, both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast Fourier transforms. Theoretically, we show that the spectra of the resulting preconditioned matrices are clustered around one. Thus Krylov subspace methods with the proposed preconditioner converge very fast. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner and show that its performance is better than the other testing preconditioners.