Stochastic collocation with kernel density estimation

The stochastic collocation method has recently received much attention for solving partial differential equations posed with uncertainty, i.e., where coefficients in the differential operator, boundary terms or right-hand sides are random fields. Recent work has led to the formulation of an adaptive collocation method that is capable of accurately approximating functions with discontinuities and steep gradients. These methods, however, usually depend on an assumption that the random variables involved in expressing the uncertainty are independent with marginal probability distributions that are known explicitly. In this work we combine the adaptive collocation technique with kernel density estimation to approximate the statistics of the solution when the joint distribution of the random variables is unknown.