Seminar in Numerical Analysis: Martin Eigel (WIAS Berlin)

Aspects of Adaptive Galerkin FEM for random PDEs

The Stochastic Galerkin FEM (SGFEM) is a common method to numerically solve PDEs with random data with the aim to obtain a functional representation of the stochastic solution. As with any spectral method, the curse of dimensionality renders the approach very challenging whenever the randomness depends on a large or even infinite set of parameters. This makes function space adaptation and model reduction strategies a necessity. We review adaptive SGFEM based on reliable a posteriori error estimators for the affine and the lognormal cases. As an alternative to a sparse discretisation, the representation in a hierarchical tensor format is examined. Moreover, as an application of the result, we present an adaptive method for explicit sampling-free Bayesian inversion.

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