An efficient iterative method for solving parameter-dependent and random diffusion problems

This paper develops and analyzes a general iterative framework for solving parameter-dependent and random diffusion problems. It is inspired by the multi-modes method of [7,8] and the ensemble method of [19] and extends those methods into a more general and unified framework. The main idea of the framework is to reformulate the underlying problem into another problem with a parameter-independent diffusion coefficient and a parameter-dependent (and solution-dependent) right-hand side, a fixed-point iteration is then employed to compute the solution of the reformulated problem. The main benefit of the proposed approach is that an efficient direct solver and a block Krylov subspace iterative solver can be used at each iteration, allowing to reuse the $LU$ matrix factorization or to do an efficient matrix-matrix multiplication for all parameters, which in turn results in significant computation saving. Convergence and rates of convergence are established for the iterative method both at the variational continuous level and at the finite element discrete level under some structure conditions. Several strategies for establishing reformulations of parameter-dependent and random diffusion problems are proposed and their computational complexity is analyzed. Several 1-D and 2-D numerical experiments are also provided to demonstrate the efficiency of the proposed iterative method and to validate the theoretical convergence results.