Adaptive Concepts for Stochastic Partial Differential Equations

An adaptive time stepping method to numerically provide approximate weak approximations of solutions of general SPDEs is proposed, where local step sizes are chosen in regard of the distance between empirical laws of subsequent time iterates and extrapolated data. The histogram-based estimator uses a data-driven partitioning of the high-dimensional state space, and efficient sampling by bootstrapping. Time adaptivity is then complemented by a local refinement/coarsening strategy of the spatial mesh of a stochastic version of the ZZ-estimator. Next to an improved accuracy, we observe a significantly reduced empirical variance of standard estimators, and therefore a reduced sampling effort. The performance of the adaptive strategies is studied for SPDEs with linear drift, including the convection-dominated case where the streamline diffusion method is adopted to attain a stable discretization, and the stochastic version of the non-linear harmonic map heat flow to the sphere \({\mathbb {S}}^{2}\) where approximate solutions exhibit discrete blow-up dynamics.