Accurate and efficient evaluation of failure probability for partial different equations with random input data

Several computational challenges arise when evaluating the failure probability of a given system in the context of risk prediction or reliability analysis. When the dimension of the uncertainties becomes high, well established direct numerical methods can not be employed because of the “curse-of-dimensionality”. Many surrogate models have been proposed with the aim of reducing computational effort. However, most of them fail in computing an accurate failure probability when the limit state surface defined by the failure event in the probability space lacks smoothness. In addition, for a stochastic system modeled by partial differential equations (PDEs) with random input, only a limited number of the underlying PDEs (order of a few tens) are affordable to solve in practice due to the considerable computational effort, therefore preventing the application of many numerical methods especially for high dimensional random inputs. In this work we develop hybrid and goal-oriented adaptive reduced basis methods to tackle these challenges by accurately and efficiently computing the failure probability of a stochastic PDE. The curse-of-dimensionality is significantly alleviated by reduced basis approximation whose bases are constructed by goal-oriented adaptation. Moreover, an accurate evaluation of the failure probability for PDE system with solution of low regularity in probability space is guaranteed by the certified a posteriori error bound for the output approximation error. At the end of this paper we suitably extend our proposed method to deal with more general PDE models. Finally we perform several numerical experiments to illustrate its computational accuracy and efficiency.