CONSTRUCTION OF PRECONDITIONERS FOR WIENER-HOPF EQUATIONS BY OPERATORSPLITTING

In this paper, we propose a new type of preconditioners for solving fi nite section Wiener-Hopf integral equations (alpha I + A(tau))x(tau) = g by the preconditioned conjugate gradient algorithm. We show that fo r an integer u > 1, the operator alpha I + A(tau) can be decomposed in to a sum of operators alpha I + P-tau((u,v)) for 0 less than or equal to v < u. Here P-tau((u,v)) are {omega(v)}-circulant integral operator s that are the continuous analog of {omega(v)}-circulant matrices. For u greater than or equal to 1, our preconditioners are defined as (1/u ) Sigma(v)(alpha I + P-tau((u,v)))(-1). Thus the way the preconditione rs are constructed is very similar to the approach used in the additiv e Schwarz method for elliptic problems. As for the convergence rate, w e prove that the spectra of the resulting preconditioned operators [(1 /u) Sigma(v)(alpha I + P-tau((u,v)))(-1)] [alpha I + A(tau)] are clust ered around 1 and thus the algorithm converges sufficiently fast. Fina lly, we discretize the resulting preconditioned equations by rectangul ar rule. Numerical results show that our methods converges faster than those preconditioned by using circulant integral operators.