Parallel domain decomposition methods for stochastic partial differential equations and analysis of nonlinear integral equations

This thesis has two parts for two different topics. Part 1 (chapter 1) is devoted to the development of parallel domain decomposition preconditioned recycling Krylov subspace methods for stochastic partial differential equations. Part 2 (chapter 2) of the thesis concerns the analysis of partial differential equations (PDEs) and singular integral equations (IEs).; In Part 1, we present parallel Schwarz type domain decomposition preconditioned recycling Krylov subspace methods for the numerical solution of stochastic elliptic problems. Karhunen-Loeve (KL) expansion and double orthogonal polynomials are used to reformulate the stochastic elliptic problems into a large number of related, but uncoupled deterministic equations. The key to an efficient algorithm lies in "recycling computed subspaces". Based on a careful analysis of the KL expansion we propose and test a grouping algorithm that tells us when to recycle and when to recompute some components of the expensive computation. We show theoretically and experimentally that the Schwarz preconditioned recycling GMRES method is optimal for the entire family of linear systems. A fully parallel implementation is provided and scalability results are reported in the thesis.; In Part 2, we consider the Euler-Lagrange systems associated with Hardy-Littlewood-Sobolev inequalities, which are the most important ingredients in the study of Sobolev spaces and nonlinear elliptic and parabolic PDEs. We prove the symmetry and monotonicity of the solutions to the Euler-Lagrange systems. We also find the optimal integrability interval for the solutions.