An adaptive sparse grid method for elliptic PDEs with stochastic coefficients

The stochastic collocation method based on the anisotropic sparse grid has become a significant tool to solve partial differential equations with stochastic inputs. The aim is to seek a vector weight and a convenient level for the method. The classical approach uses a posteriori approach on the solution, yielding to an additional prohibitive cost with a large stochastic dimension. In this work, we discuss an adaptive approach of this method to calculate the statistics of the solution. It is based on an adaptive approximation of the inverse diffusion parameter. We construct an efficient error indicator which is an upper bound of the error on the solution. In the case of unbounded variables, we use an appropriate error estimation to compute suitable weights of the method. Numerical examples are presented to confirm the efficiency of the approach and showing that the cost is considerably reduced without loss of accuracy.