Towards a unified multiresolution scheme for treating discontinuities in differential equations with uncertainties

In the present work, a method for solving partial differential equations with uncertainties is presented. A multiresolution method, permitting to compute statistics for the entire solution and in presence of a whatever form of the probability density function, is extended to perform an adaptation in both physical and stochastic spaces. The efficiency of this strategy, in terms of refinement/coarsening capabilities, is demonstrated on several test-cases by comparing with respect to other more classical techniques, namely Monte Carlo (MC) and Polynomial Chaos (PC). Finally, the proposed strategy is applied to the heat equation showing very promising results in terms of accuracy, convergence and regularity.