Higher-order quadratures for circulant preconditioned Wiener-Hopf equations

Abstract

In this paper, we consider solving matrix systems arising from the discretization of Wiener-Hopf equations by preconditioned conjugate gradient (PCG) methods. Circulant integral operators as preconditioners have been proposed and studied. However, the discretization of these preconditioned equations by employing higher-order quadratures leads to matrix systems that cannot be solved efficiently by using fast Fourier transforms (FFTs). The aim of this paper is to propose new preconditioners for Wiener-Hopf equations. The discretization of these preconditioned operator equations by higher-order quadratures leads to matrix systems that involve only Toeplitz, circulant and diagonal matrix-vector multiplications and hence can be computed efficiently by FFTs in each iteration. We show that with the proper choice of kernel functions of Wiener-Hopf equations, the resulting preconditioned operators will have clustered spectra and therefore the PCG method converges very fast. Numerical examples are given to illustrate the fast convergence of the method and the improvement of the accuracy of the computed solutions with using higher-order quadratures.