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2024 | OriginalPaper | Chapter

A-Posteriori QMC-FEM Error Estimation for Bayesian Inversion and Optimal Control with Entropic Risk Measure

Authors : Marcello Longo, Christoph Schwab, Andreas Stein

Published in: Monte Carlo and Quasi-Monte Carlo Methods

Publisher: Springer International Publishing

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Abstract

We propose a novel a-posteriori error estimation technique where the target quantities of interest are ratios of high-dimensional integrals, as occur e.g. in PDE constrained Bayesian inversion and PDE constrained optimal control subject to an entropic risk measure. We consider in particular parametric, elliptic PDEs with affine-parametric diffusion coefficient, on high-dimensional parameter spaces. We combine our recent a-posteriori Quasi-Monte Carlo (QMC) error analysis, with Finite Element a-posteriori error estimation. The proposed approach yields a computable a-posteriori estimator which is reliable, up to higher order terms. The estimator’s reliability is uniform with respect to the PDE discretization, and robust with respect to the parametric dimension of the uncertain PDE input.

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previous chapter Comparison of Two Search Criteria for Lattice-Based Kernel Approximation

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In practice, h either parametrizes the local mesh-size \( \max _{T\in {\mathcal {T}}_h} |T|^{1/2} \), for quasi-uniform collections of partitions, or it relates to the refinement level in case of adaptive refinement [16].

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Title: A-Posteriori QMC-FEM Error Estimation for Bayesian Inversion and Optimal Control with Entropic Risk Measure
Authors: Marcello Longo
Christoph Schwab
Andreas Stein
Publisher: Springer International Publishing
Book: Monte Carlo and Quasi-Monte Carlo Methods
Print ISBN: 978-3-031-59761-9

Electronic ISBN: 978-3-031-59762-6

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-59762-6_21

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