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 un can be determined in O(n★) flops and the accuracy ∥u-un∥0,b⩽cn★-m∥f(m)∥0,a is achieved where n★=n★(n) is the number of sample points at which the values of f and K, the kernel of the integral operator, are involved; moreover, we require that, having determined the parameters of un, the value of un at any particular point x∈(-∞,∞) is available with the same accuracy O(n★-m) at the cost of O(1) flops. Here ∥·∥0,a and ∥·∥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.