Fast numerical collocation solutions of integral equations

We present a numerical implementation of a fast multiscale collocation method for solving Fredholm integral equations of the second kind with weakly singular kernels. The general setting of such a collocation method was recently developed by Chen, Micchelli and Xu. Following the general setting, in this paper we consider three important problems for the practical use of such collocation methods. The first problem regards the construction of concrete multiscale piecewise linear, quadratic and cubic polynomial functions and the corresponding multiscale collocation functionals. The second problem that we address is the practical truncation of the coefficient matrix. We propose a  block truncation strategy which allows us to compress the matrix without computing the distances between the supports of a basis function and a collocation functional. The last problem is the fast numerical solution of the large discrete linear system resulting from the compression. We make use of the multiscale structure and the sparseness of the coefficient matrix in developing fast solver for the linear system. Numerical examples are presented to demonstrate the accuracy and computational speed of the method.