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

[摘要] This study considers the solution of a class of linear systems related with the fractionalPoisson equation (FPE)(−∇2)α/2φ=g(x,y)with nonhomogeneous boundary conditions on abounded domain. A numerical approximation to FPE is derived using a matrix representation of theLaplacian to generate a linear system of equations with its matrixAraised to the fractional powerα/2. The solution of the linear system then requires the action of the matrix functionf(A)=A−α/2on a vectorb. For large, sparse, and symmetric positive definite matrices, the Lanczos approximationgeneratesf(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade ofAwith respecttoband the residual for the linear system are sufficiently small. Memory constraints oftenrequire restarting the Lanczos decomposition; however this is not straightforward in the context ofmatrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioningfor solving linear systems to improve convergence of the Lanczos approximation. Wegive an error bound for the new method and illustrate its role in solving FPE. Numerical results areprovided to gauge the performance of the proposed method relative to exact analytic solutions.

[学科分类] 应用数学