A Simpler GMRES Method for Oscillatory Integrals with Irregular Oscillations

Qinghua Wu,Meiying Xiang

doi:10.1155/2015/103074

Abstract

A simpler GMRES method for computing oscillatory integral is presented. Theoretical analysis shows that this method is mathematically equivalent to the GMRES method proposed by Olver (2009). Moreover, the simpler GMRES does not require upper Hessenberg matrix factorization, which leads to much simpler program and requires less work. Numerical experiments are conducted to illustrate the performance of the new method and show that in some cases the simpler GMRES method could achieve higher accuracy than GMRES.

Highlights

IntroductionIn this paper we consider iterative methods for computing high oscillatory integral b
In this paper we consider iterative methods for computing high oscillatory integral b∫ f (x) eiwg(x)dx, (1)a where f(x) and g(x) are smooth functions and g󸀠(x) ≠ 0 for x ∈ (a, b) and nonoscillatory with respect to increasing w
We look for an orthogonal basis Pn = (p1, . . . , pn) of DKn(D, f), where problem (3) is reduced to an upper triangular least-squares problem. We show that this method is mathematically equivalent to the GMRES method and has some other properties as those possessed by GMRES but requires less work

Summary

Introduction

In this paper we consider iterative methods for computing high oscillatory integral b. In the last few years many efficient methods have been devised for approximating this kind of oscillatory integrals, such as asymptotic method [1], Filon-type method [2], Levin’s collocation method [3], modified Clenshaw-Curtis method, Clenshaw-Curtis-Filon-type method [4], generalized quadrature rule [5], and numerical steepest descent method [6]. In many situations the accuracy of the Filon-type method is significantly higher than that of the asymptotic method, even though it is of the same order To work around this weakness, Xiang [7] derived efficient Filon-type method, an approach without computing the moments. Numerical steepest descent method achieves a higher asymptotic order than any other method It requires the integrand being analytic and deforming the path of integration into the complex plane, which in practice add complexity to the method.

Simpler GMRES for Differentiation Operator

Convergence Rate

Numerical Examples

Conclusion