Journal of Applied Mathematics and Stochastic Analysis
Volume 2008 (2008), Article ID 104525, 26 pages
Research Article

A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

M. Ilić, I. W. Turner, and V. Anh

School of Mathematical Sciences, Queensland University of Technology, Qld 4001, Australia

Received 21 May 2008; Revised 10 September 2008; Accepted 23 October 2008

Academic Editor: Nikolai Leonenko

Copyright © 2008 M. Ilić et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=Aα/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)bβ0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.