A Multilevel Monte Carlo Method for Computing Failure Probabilities

We propose and analyze a method for computing failure probabilities of systems modeled as numerical deterministic models (e.g., PDEs) with uncertain input data. A failure occurs when a functional of the solution to the model is below (or above) some critical value. By combining recent results on quantile estimation and the multilevel Monte Carlo method, we develop a method that reduces computational cost without loss of accuracy. We show how the computational cost of the method relates to error tolerance of the failure probability. For a wide and common class of problems, the computational cost is asymptotically proportional to solving a single accurate realization of the numerical model, i.e., independent of the number of samples. Significant reductions in computational cost are also observed in numerical experiments

Uncertainty Quantification for Approximate p-Quantiles for Physical Models with Stochastic Inputs

We consider the problem of estimating the p-quantile for a given functional evaluated on solutions of a deterministic model in which model input is subject to stochastic variation. We derive upper and lower bounding estimators of the p-quantile. We perform an a posteriori error analysis for the p-quantile estimators that takes into account the effects of both the stochastic sampling error and the deterministic numerical solution error and yields a computational error bound for the estimators. We also analyze the asymptotic convergence properties of the p-quantile estimator bounds in the limit of large sample size and decreasing numerical error and describe algorithms for computing an estimator of the p-quantile with a desired accuracy in a computationally efficient fashion. One algorithm exploits the fact that the accuracy of only a subset of sample values significantly affects the accuracy of a p-quantile estimator resulting in a significant gain in computational efficiency. We conclude with a number of numerical examples, including an application to Darcy flow in porous media

3rd International Congress on Arsenic in the Environment (AS 2010): Arsenic in Geosphere and Human Diseases

Until recently, most estimates of arsenic (As) pollution have focused on the predominance of As poisoning in the groundwater of West Bengal (India) and Bangladesh (Ahmed et al., 2004), which has been thought to be limited to the Ganges Delta (the lower Ganges Plain). Several authors suggested that the reductive dissolution of Fe(Ill)-­ oxyhydroxides in strongly reducing conditions of the young alluvial sediments is the cause for As mobilization (Bhattacharya et al., 1997; Ahmed et al., 2004; McArthur et al., 2001; Nickson el al., 1998; von Bromssen et al., 2007). Holocene alluvial aquifers of Ballia, Ghazipur and Bhagalpur district in the middle Gangetic Plain have high concentra­tions of geogenic As. The As contaminated aquifers are pervasive within lowland organic rich, clayey deltaic sedi­ments in the Bengal Basin and locally within simi­lar facies in narrow, entrenched river valleys within the Ganges Alluvial Plain (Acharyya & Shah, 2004; Mukherjee et al., 2006). This study has been carried out with the following objectives: I) to quantify the As in the groundwater of the Ghazipur District, Uttar Pradesh, India, and 2) to understand the mechanism controlling the mobilization of As and its evolution