  We propose algorithms for solving high-dimensional Partial Differential
Equations (PDEs) that combine a probabilistic interpretation of PDEs, through
Feynman-Kac representation, with sparse interpolation. Monte-Carlo methods and
time-integration schemes are used to estimate pointwise evaluations of the
solution of a PDE. We use a sequential control variates algorithm, where
control variates are constructed based on successive approximations of the
solution of the PDE. Two different algorithms are proposed, combining in
different ways the sequential control variates algorithm and adaptive sparse
interpolation. Numerical examples will illustrate the behavior of these
algorithms.
