Construction of Preconditioners for Wiener-Hopf Equations by Operator Splitting

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 {ω_v}-circulant integral operators that are the continuous analog of {ω_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.