Wavelet Monte Carlo Methods for the Global Solution of Integral Equations

We study the global solution of Fredholm integral equations of the second kind by the help of Monte Carlo methods. Global solution means that we seek to approximate the full solution function. This is opposed to the usual applications of Monte Carlo, were one only wants to approximate a functional of the solution. In recent years several researchers developed Monte Carlo methods also for the global problem. In this paper we present a new Monte Carlo algorithm for the global solution of integral equations. We use multiwavelet expansions to approximate the solution. We study the behaviour of variance on increasing levels, and based on this, develop a new variance reduction technique. For classes of smooth kernels and right hand sides we determine the convergence rate of this algorithm and show that it is higher than those of previously developed algorithms for the global problem. Moreover, an information-based complexity analysis shows that our algorithm is optimal among all stochastic algorithms of the same computational cost and that no deterministic algorithm of the same cost can reach its convergence rate.