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We study the weak approximation problem of diffusions, which are reflected at a subset of the boundary of a domain and stopped at the remaining boundary. First, we derive an error representation for the projected Euler method of Costantini, Pacchiarotti and Sartoretto [Costantini et al., SIAM J. Appl. Math., 58(1):73–102, 1998], based on which we introduce two new algorithms. The first one uses a correction term from the representation in order to obtain a higher order of convergence, but the computation of the correction term is, in general, not feasible in dimensions d > 1. The second algorithm is adaptive in the sense of Moon, Szepessy, Tempone and Zouraris [Moon et al., Stoch. Anal. Appl., 23:511–558, 2005], using stochastic refinement of the time grid based on a computable error expansion derived from the representation. Regarding the stopped diffusion, it is based in the adaptive algorithm for purely stopped diffusions presented in Dzougoutov, Moon, von Schwerin, Szepessy and Tempone [Dzougoutov et al., Lect. Notes Comput. Sci. Eng., 44, 59–88, 2005]. We give numerical examples underlining the theoretical results.

We present a systematic study of efficient sampling methods to quantify parametric uncertainty in stage II crack propagation and in a Burgers equation with random viscosity. The samplingmethods employed are Monte Carlo and quasi-Monte Carlo methods enhanced through variance reduction techniques including stratified sampling. We also present a recent variance reduction technique called sensitivity derivative enhanced sampling that utilizes sensitivity derivatives to make more efficient use of random samples. The crack propagation problem is a benchmark problem proposed by the Society of Automotive Engineers (SAE); this problem was selected because its large variance can hinder the already slow convergence of typical Monte Carlo methods and thus provides an ideal test for efficient sampling methods. We also examine a steady-state Burgers equation, a model equation of perennial interest for its connections with the equations of fluid mechanics, with uncertain viscosity. We demonstrate that sensitivity derivative enhanced sampling (SDES) can reduce the required number of samples by an order of magnitude to achieve the same accuracy as a traditionalMonte Carlo method. When SDES is coupled with other sampling techniques such as stratified sampling, the number of samples required can be reduced by as much as two orders of magnitude. In addition, we introduce an analogous approach that utilizes sensitivity derivatives which, under appropriate conditions, results in a more efficient quasi-Monte Carlo method.