PRECONDITIONERS FOR WIENER-HOPF EQUATIONS WITH HIGH-ORDER QUADRATURE-RULES

We consider solving the Wiener-Hopf equations with high-order quadratu re rules by preconditioned conjugate gradient (PCG) methods. We propos e using convolution operators as preconditioners for these equations. We will show that with the proper choice of kernel functions for the p reconditioners, the resulting preconditioned equations will have clust ered spectra and therefore can be solved by the PCG method with superl inear convergence rate. Moreover, the discretization of these equation s 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 illust rate the fast convergence of the method and the improvement on accurac y by using higher-order quadrature rule. We also compare the performan ce of our preconditioners with the circulant integral operators.