A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation
Joint Authors
Anh, V. V.
Ilić, M.
Turner, I. W.
Source
Journal of Applied Mathematics and Stochastic Analysis
Issue
Vol. 2008, Issue 2008 (31 Dec. 2008), pp.1-26, 26 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2009-03-11
Country of Publication
Egypt
No. of Pages
26
Main Subjects
Abstract EN
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)=A−α/2 on a vector b.
For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(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.
American Psychological Association (APA)
Ilić, M.& Turner, I. W.& Anh, V. V.. 2009. A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation. Journal of Applied Mathematics and Stochastic Analysis،Vol. 2008, no. 2008, pp.1-26.
https://search.emarefa.net/detail/BIM-446682
Modern Language Association (MLA)
Ilić, M.…[et al.]. A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation. Journal of Applied Mathematics and Stochastic Analysis No. 2008 (2008), pp.1-26.
https://search.emarefa.net/detail/BIM-446682
American Medical Association (AMA)
Ilić, M.& Turner, I. W.& Anh, V. V.. A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation. Journal of Applied Mathematics and Stochastic Analysis. 2009. Vol. 2008, no. 2008, pp.1-26.
https://search.emarefa.net/detail/BIM-446682
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-446682