An adaptive sparse grid method for elliptic PDEs with stochastic coefficients

The stochastic collocation method based on an anisotropic sparse grid is nowadays a significant tool to solve partial differential equations with random input data. This method is based on a level of interpolation and weights of anisotropy. The objective of some adaptive approaches is to select cleverly these parameters, in order to reduce the computational cost.In this work, we propose such an adaptive approach, based on an approximation of the inverse diffusion coefficient. We introduce an error indicator which is an upper bound of the error in the solution and use this indicator as a reliable and cheap tool for choosing the level of interpolation. We also propose a new error estimation in one dimension, for unbounded random variables, and use it to compute suitable weights.Numerical examples show the efficiency of our methodology, since the cost is considerably reduced, without loss of accuracy.