Preconditioning techniques for stochastic partial differential equations

Abstract

This thesis is about preconditioning techniques for time dependent stochastic Partial Differential Equations arising in the broader context of Uncertainty Quantification. State-of-the-art methods for an efficient integration of stochastic PDEs require the solution field to lie on a low dimensional linear manifold. In cases when there is not such an intrinsic low rank structure we must resort on expensive and time consuming simulations. We provide a preconditioning technique based on local time stretching capable to either push or keep the solution field on a low rank manifold with substantial reduction in the storage and the computational burden. As a by-product we end up addressing also classical issues related to long time integration of stochastic PDEs.