关于非局部扩散模型的一种快速预处理算法

A Fast Preconditioned Algorithm for Nonlocal Diffusion Model

摘要:

变系数非局部扩散模型可以被一种快速配置法进行有效的数值离散。离散后得到一个系数矩阵具有 Toeplitz 结构且稠密的线性方程组。由于系数矩阵是非对称的，该线性方程组可以用广义极小残量法（GMRES）方法求解。为了提高 GMRES 方法的收敛率，构造了系数矩阵的 Toeplitz 及循环预处理子，并提出了预处理 GMRES 方法求解该线性方程组。数值算例也表明了该预处理算法的有效性。

Abstract:

A fast collocation scheme can be used to discretize the variable-coef?cient nonlocal diffusion model effectively. The coefficient matrix of the resulting linear system is unsymmetrical， dense and Toeplitz-like. The generalized minimum residual （GMRES） method can be employed to solve the discretized linear systems. In order to improve the rate of convergence of the GMRES method， the Toeplitz preconditioner and circulant preconditioner are constructed for the coefficient matrix， and the preconditioned GMRES methods are proposed for solving the discretized linear systems. Numerical examples are presented to illustrate the effectiveness of the preconditioned methods.