A posteriori error analysis and adaptivity for high-dimensional elliptic and parabolic boundary value problems

Abstract

We derive a posteriori error estimates for the (stopped) weak Euler method to discretize SDE systems which emerge from the probabilistic reformulation of elliptic and parabolic (initial) boundary value problems. The a posteriori estimate exploits the use of a scaled random walk to represent noise, and distinguishes between realizations in the interior of the domain and those close to the boundary. We verify an optimal rate of (weak) convergence for the a posteriori error estimate on deterministic meshes. Based on this estimate, we then set up an adaptive method which automatically selects local deterministic mesh sizes, and prove its optimal convergence in terms of given tolerances. Provided with this theoretical backup, and since corresponding Monte-Carlo based realizations are simple to implement, these methods may serve to efficiently approximate solutions of high-dimensional (initial-)boundary value problems.