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A posteriori error estimation and adaptivity for stochastic collocation FEM

This talk concerns the design and analysis of an adaptive FEM-based solution strategy for PDE problems with uncertain or parameter-dependent inputs. High-fidelity numerical solutions to these problems are infeasible to compute in many realistic applications. Conversely, so-called surrogate approximations that are functions of the (stochastic) parameters are not only effective in approximating the input-output map but can also be used to estimate a wide range of quantities of interest—specific (and usually localised) features of the solution to a PDE. The talk will address: (i) the reliable control of errors in computing solutions to parametric PDEs (or the associated quantities of interest) using surrogate approximations, and (ii) systematic approaches to reduce those errors via adaptive algorithms.

We will focus on one particular type of surrogate approximations—sparse grid stochastic collocation finite element approximations that combine sparse grid interpolation in the parameter domain with finite element discretisation in the physical (spatial) domain. We propose a novel approach to a posteriori error estimation in this approximation setting. This employs hierarchical error estimates that are useful for reliable error control and also provide practical error indicators for guiding the adaptive refinement process. We will demonstrate the effectivity and robustness of the proposed error estimation strategy. We will then discuss the performance of the developed adaptive algorithms. The latter include goal-oriented adaptive algorithms for approximating quantities of interest represented by linear or nonlinear goal functionals.

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