Toward a Unified Multiresolution Scheme in the Combined Physical/Stochastic Space for Stochastic Differential Equations

In the present work, an innovative method for solving stochastic partial differential equations is presented. A multiresolution method permitting to compute statistics of the quantity of interest for a whatever form of the probability density function is extended to permit an adaptation in both physical and stochastic spaces. The efficiency of this strategy, in terms of refinement/derefinement capabilities, is displayed for stochastic algebraic and differential equations with respect to other more classical techniques, like Monte Carlo (MC) and Polynomial Chaos (PC). Finally, the proposed strategy is applied to the heat equation, displaying very promising results in terms of accuracy, convergence and regularity