Goal-oriented adaptive algorithms for stochastic collocation finite element methods

Round, Thomas (2024). Goal-oriented adaptive algorithms for stochastic collocation finite element methods. University of Birmingham. Ph.D.

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Abstract

In this thesis, our main objective is the numerical approximation of linear quantities of interest for problems involving elliptic partial differential equations (PDEs) which admit inputs with either an affine or non-affine dependence on random parameters. Our approach will involve a stochastic collocation finite element method, which is based on a deterministic finite element method (FEM), combined with a sparse grid collocation procedure. This method will be introduced in the context of adaptive algorithms, which are furnished by a posteriori error estimation techniques that are designed to independently influence both spatial and parametric adaptivity.

Firstly, we produce new results in the stochastic collocation FEM setting that reveal the impact of different strategic choices in the context of an adaptive algorithm. Specifically, we investigate choices of hierarchical spatial error estimation strategy, the use of different families of collocation points within the construction of sparse grids, and the use of the reduced margin (in comparison to the full margin) for the purposes of enriching the underlying sparse grid.

Secondly, we propose an error estimation strategy for the goal-oriented framework which utilises products of hierarchical estimators in the stochastic collocation FEM setting. This involves the use of a novel correction term to compensate for the lack of global Galerkin orthogonality in this setting. Our error estimation strategy drives a goal-oriented adaptive algorithm with innovative marking procedures to simultaneously handle the interplay between primal and dual, as well as spatial and parametric contributions to our error estimate. We provide an upper bound for the underlying error estimate, and demonstrate the performance of the resulting adaptive algorithm in extensive numerical experiments.

Thirdly, we suggest a dual-weighted residual estimate for the error in the quantity of interest, and a variant of this which utilises the primal problem as weightings for residuals associated with the dual problem. We prove the reliability of both the standard dual-weighted residual method, and the new symmetric dual-weighted residual, for the purposes of estimating the error in the quantity of interest. Further numerical experiments illustrate the performance of the corresponding adaptive algorithm.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):