Strong convergence analysis of iterative solvers for random operator equations

For the numerical solution of linear systems that arise from discretized linear partial differential equations, multigrid and domain decomposition methods are well established. Multigrid methods are known to have optimal complexity and domain decomposition methods are in particular useful for parallelization of the implemented algorithm. For linear random operator equations, the classical theory is not directly applicable, since condition numbers of system matrices may be close to degenerate due to non-uniform random input. It is shown that iterative methods converge in the strong, i.e. Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}, sense if the random input satisfies certain integrability conditions. As a main result, standard multigrid and domain decomposition methods are applicable in the case of linear elliptic partial differential equations with lognormal diffusion coefficients and converge strongly with deterministic bounds on the computational work which are essentially optimal. This enables the application of multilevel Monte Carlo methods with rigorous, deterministic bounds on the computational work.