Fast solvers of integral equations of the second kind: wavelet methods

For the Fredholm integral equation u = Tu + f on the real line, fast solvers are designed on the basis of a discretized wavelet Galerkin method with the Sloan improvement of the Galerkin solution. The Galerkin system is solved by GMRES or by the Gauss elimination method. Our concept of the fast solver includes the requirements that the parameters of the approximate solution u(n) can be determined O(n(*)) flops and the accuracy parallel to u - u(n)parallel to(0,b) <= cn(*)(-m)parallel to f((m))parallel to(0),(a) is achieved where n(*) = n(*)(n) is the number of sample points at which the values off and K, the kernel of the integral operator, are involved; moreover, we require that, having determined the parameters Of un, the value Of u(n) at any particular point x is an element of (-infinity, infinity) is available with the same accuracy 0(n* m) at the cost of 0(l) flops. Here parallel to (.) parallel to(0,a) and parallel to (.) parallel to(0,b) are certain weighted uniform norms. Using GMRES, the 2m-smoothness of K is sufficient; in case of Gauss method, K must be smoother. (c) 2004 Elsevier Inc. All rights reserved.