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“Adaptive methods for stochastic PDEs” with Prof. Andreas Prohl, Univ. of Tübingen
The construction of adaptive methods to approximate solutions of deterministic PDEs is an established part in the numerical analysis of PDEs: many of them exploit a posteriori error estimates – where numerical errors are bounded by computable terms only – to steer automatic space(-time) mesh refinements/coarsenings to accomplish the aim to accurately represent approximate solutions on meshes as coarse as possible.
A corresponding strategy is relevant as well for stochastic PDEs, where the numerical analysis is much less developed. Immediate questions appear: what type of meshes should be given preference (‘deterministic or random’), or what are relevant errors (‘weak or strong errors, or errors in law’) on which to base automatic space-time remeshing?
I start this series of talks with an a priori error analysis of the stochastic heat equation, repeating the practically relevant variational solution concept for this SPDE, of stability estimates for its (time-implicit) spatio-temporal discretization, and of Kolmogorov’s equation. The main part then proposes three conceptionally different adaptive methods, whose relevancy depends on the answer to my question above:
(i)tthis method focuses on a strong approximation, which is based on residual-based estimators.
(ii)there we aim for a weak approximation, which uses the related Kolmogorov’s equation.
(iii)tthis method is based on a comparison of subsequent empirical measures, which are constructed via data-dependent partitionings of the high-dimensional finite-element space.
The lectures base on joint works with A.K. Majee (IIT Delhi), with C. Schellnegger (earlier: U Tuebingen), and F. Merle (U Tuebingen).