NONPARAMETRIC DENSITY ESTIMATION FOR RANDOMLY PERTURBED ELLIPTIC PROBLEMS I: COMPUTATIONAL METHODS, A POSTERIORI ANALYSIS, AND ADAPTIVE ERROR CONTROL D. ESTEP * , A. M ˚ ALQVIST † , AND S. TAVENER ‡ Abstract. We consider the nonparametric density estimation problem for a quantity of interest computed from solutions of an elliptic partial differential equation with randomly perturbed coef- ficients 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 Neu- mann 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. Key words. a posteriori error analysis, adjoint problem, density estimation, domain decom- position, elliptic problem, Neumann series, nonparametric density estimation, random perturbation, sensitivity analysis AMS subject classifications. 65N15, 65N30, 65N55, 65C05 1. Introduction. The practical application of differential equations to model physical phenomena presents problems in both computational mathematics and statis- tics. The mathematical issues arise because of the need to compute approximate so- lutions of difficult problems while statistics arises because of the need to incorporate experimental data and model uncertainty. The consequence is that significant error and uncertainty attends any computed information from a model applied to a con- crete situation. The problem of quantifying that error and uncertainty is critically important. We consider the nonparametric density estimation problem for a quantity of in- terest computed from the solutions of an elliptic partial differential equation with randomly perturbed coefficients and data. The ideal problem is to compute a quan- tity of interest Q(U ), expressed as a linear functional, of the solution U of  −∇ · ( A(x)∇U ) = G(x), x ∈ Ω, U =0, x ∈ ∂ Ω, (1.1) where Ω is a convex polygonal domain with boundary ∂ Ω and A(x),G(x) are stochas- tic functions that vary randomly according to some given probability structure. The * Department of Mathematics and Department of Statistics, Colorado State University, Fort Collins, CO 80523 (estep@math.colostate.edu). D. Estep’s work is supported in part by the De- partment of Energy (DE-FG02-04ER25620, DE-FG02-05ER25699, DE-FC02-07ER54909), Lawrence Livermore National Laboratory (B573139), the National Aeronautics and Space Administration (NNG04GH63G), the National Science Foundation (DMS-0107832, DMS-0715135, DGE-0221595003, MSPA-CSE-0434354, ECCS-0700559), Idaho National Laboratory (00069249), and the Sandia Cor- poration (PO299784) † Department of Information Technology, Uppsala University, SE-751 05 Uppsala, Sweden (axel.malqvist@it.uu.se. A. M˚ alqvist was supported in part by the Department of Energy (DE- FG02-04ER25620)) ‡ Department of Mathematics, Colorado State University, Fort Collins, CO 80523 (tavener@math.colostate.edu). S. Tavener’s work is supported in part by the Department of Energy (DE-FG02-04ER25620) 1