It is well known that there is a mathematical equivalence between "solving" parabolic partial differential equations and "the integration" of certain functionals on Wiener Space. Monte Carlo simulation of stochastic differential equations is a naive approach based on this underlying principle.
In one dimension, it is well known that Gaussian quadrature can be a very effective approach to integration. We discuss the appropriate extension of this idea to Wiener Space. In the process we develop high order numerical schemes valid for high dimensional SDEs and semi-elliptic PDEs.