Construction of preconditioners for Wiener-Hopf equations by operator splitting

Abstract

In this paper, we propose a new type of preconditioners for solving finite section Wiener-Hopf integral equations ( αI + A τ ) x τ = g by the preconditioned conjugate gradient algorithm. We show that for an integer u > 1, the operator αI + A τ > can be decomposed into a sum of operators αI + P τ ( u, v) for 0 ≤ v < u. Here P τ ( u, v) are gw v circulant matrices. For u - 1, our preconditioners are defined as ( 1 u )∑ v(αI+P τ (u,v)) −1 . Thus the way the preconditioners are constructed is very similar to the approach used in the additive Schwarz method for elliptic problems. As for the convergence rate, we prove that the spectra of the resulting preconditioned operators ( 1 u )∑ v(αI+P τ (u,v)) −1][αI+A τ are clustered around 1 and thus the algorithm converges sufficiently fast. Finally, we discretize the resulting preconditioned equations by rectangular rule. Numerical results show that our methods converges faster than those preconditioned by using circulant integral operators.