Circulant integral operators as preconditioners for Wiener-Hopf equations

In this paper, we study the solutions of finite-section Wiener-Hopf equations by the preconditioned conjugate gradient method. Our main aim is to give an easy and general scheme of constructing good circulant integral operators as preconditioners for such equations. The circulant integral operators are constructed from sequences of conjugate symmetric functions {C_τ}_τ. Let k(t) denote the kernel function of the Wiener-Hopf equation and k̂(t) be its Fourier transform. We prove that for sufficiently large τ if {C_τ}_τ is uniformly bounded on the real line R and the convolution product of the Fourier transform of C_τ with k̂(t) uniformly on R, then the circulant preconditioned Wiener-Hopf operator will have a clustered spectrum. It follows that the conjugate gradient method, when applied to solving the preconditioned operator equation, converges superlinearly. Several circulant integral operators possessing the clustering and fast convergence properties are constructed explicitly. Numerical examples are also given to demonstrate the performance of different circulant integral operators as preconditioners for Wiener-Hopf operators.