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...

In this supplement to [14], we propose a new Calderón-type preconditioner for the weakly singular integral operator for −∆ on screens in R. We introduce a modified hypersingular operator, which is the exact inverse of the weakly singular operator on the unit disk. It forms the foundation for dual-mesh-based operator preconditioning. Applied to low-order boundary element Galerkin discretizations...

Three iterative refinement schemes are studied for approximating the solutions of linear weakly singular Fredholm integral equations of the second kind. The rates of convergence and computational costs of the three schemes are studied and compared with the classical approach by applying them respectively to: (i) a sparse linear system associated with an integral equation modelling a real life A...

Questions of solution of three-dimensional diffraction problems are considered. Each problem is formulated as a single weakly-singular integral equation of the 1st kind for a single unknown function. Discretization of these equations is realized by means of special smoothing method of integral operators kernels. Numerical solutions of systems of linear algebraic equations, approximating integra...