Nonparametric Density Estimation for Randomly Perturbed Elliptic Problems Iii: Convergence, Complexity, and Generalizations

This is the last in a series of three papers on nonparametric density estimation for randomly perturbed elliptic problems. In the previous papers [3, 4] an efficient algorithm for propagation of uncertainty into a quantity of interest computed from numerical solutions of an elliptic partial differential equation was presented, analyzed, and applied to different problems in e.g. oil reservoir simulation. In this paper we focus on convergence, complexity, and generalizations. The convergence result is a new and crucial contribution. The proof is based on the assumption that the underlying domain decomposition algorithm converges geometrically. The main ideas of the proof can be applied to a large class of domain decomposition algorithms.