Monte Carlo approximation of weakly singular integral operators
We study the randomized approximation of weakly singular integral operators. For a suitable class of kernels having a standard type of singularity and being otherwise of finite smoothness, we develop a Monte Carlo multilevel method, give convergence estimates and prove lower bounds which show the optimality of this method and establish the complexity. As an application we obtain optimal methods for and the complexity of randomized solution of the Poisson equation in simple domains, when the solution is sought oil subdornains of arbitrary dimension. (c) 2005 Elsevier Inc. All rights reserved.
weakly singular integral operator lower bounds complexity monte carlo algorithm