Preconditioners for Wiener--Hopf Equations with High-Order Quadrature Rules

We consider solving the Wiener--Hopf equations with high-order quadrature rules by preconditioned conjugate gradient (PCG) methods. We propose using convolution operators as preconditioners for these equations. We will show that with the proper choice of kernel functions for the preconditioners, the resulting preconditioned equations will have clustered spectra and therefore can be solved by the PCG method with superlinear convergence rate. Moreover, the discretization of these equations by high-order quadrature rules leads to matrix systems that involve only Toeplitz or diagonal matrix--vector multiplications and hence can be computed efficiently by FFTs. Numerical results are given to illustrate the fast convergence of the method and the improvement on accuracy by using higher-order quadrature rule. We also compare the performance of our preconditioners with the circulant integral operators.