Nonparametric Density Estimation for Randomly Perturbed Elliptic Problems I: Computational Methods, A Posteriori Analysis, and Adaptive Error Control

We consider the nonparametric density estimation problem for a quantity of interest computed from solutions of an elliptic partial differential equation with randomly perturbed coefficients and data. Our particular interest are problems for which limited knowledge of the random perturbations are known. We derive an efficient method for computing samples and generating an approximate probability distribution based on Lion’s domain decomposition method and the Neumann series. We then derive an a posteriori error estimate for the computed probability distribution reflecting all sources of deterministic and statistical errors. Finally, we develop an adaptive error control algorithm based on the a posteriori estimate.