Stochastic methods for solving partial differential equations in high dimension

Standard numerical methods (such as finite elements) are efficient to solve PDEs in low dimension but intractable for high-dimensional problems. In order to overcome these limits, we propose an adaptive sparse approximation method based on a probabilistic interpretation of PDEs (using Feynman-Kac representation). Monte-Carlo methods are used to get noisy pointwise evaluations of the solution of a PDE and to construct an approximate interpolation of this solution. Here pointwise evaluations are obtained using a sequential control variates algorithm proposed by Gobet & Maire, where control variates are constructed from successive approximations of the solution of the PDE. Two different algorithms are proposed, combining adaptive sparse approximation and sequential control variates algorithm in two different ways. We will show different numerical examples to illustrate the behavior of the algorithms.