On the application of GMRES to oscillatory singular integral equations

TL;DR

This paper introduces a novel numerical method leveraging explicit formulas for Cauchy integrals and the GMRES algorithm to efficiently solve oscillatory singular integral equations, with potential applications in inverse scattering problems.

Contribution

The paper develops a new approach combining explicit integral formulas and GMRES for oscillatory singular integral equations, improving convergence and linking theory with numerical analysis.

Findings

Achieved dramatic acceleration in GMRES convergence.

Established explicit formulas for Cauchy integrals of complex exponentials.

Proposed a step towards inverse scattering transform solvers.

Abstract

We present a new method for the numerical solution of singular integral equations on the real axis. The method's value stems from an explicit formula for the Cauchy integral of a complex exponential multiplied by a rational function. Additionally, the inner product of such functions is computed explicitly. With these tools, the GMRES algorithm is applied to both non-oscillatory and oscillatory singular integral equations. Ideas from Fredholm theory and Riemann--Hilbert problems are used to motivate preconditioners for these singular integral equations. A dramatic acceleration in convergence is realized. This presents a strong link between the theory of singular integral equations and the numerical analysis of such equations. Furthermore, this method presents a first step towards a solver for the inverse scattering transform that does not require the deformation of a Riemann--Hilbert problem.