Nonparametric density estimation for randomly perturbed elliptic problems III: convergence, computational cost, and generalizations

This is the third in a series of three papers on nonparametric density estimation for randomly perturbed elliptic problems. In the previous papers by Estep, M?lqvist, and Tavener (SIAM J. Sci. Comput. 31:2935?C2959,?2009; Int. J. Numer. Methods Eng. 80:846?C867,?2009), we derive an a posteriori error estimate for a computed probability distribution and an efficient adaptive algorithm for propagation of uncertainty into a quantity of interest computed from numerical solutions of an elliptic partial differential equation. We also test the algorithm on various problems including an example relevant to oil reservoir simulation. In this paper, we derive a convergence result for the method 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. We also present several generalizations of the method and an analysis of the computational cost.