Inverse Problems in a Bayesian Setting

Hermann G. Matthies, Elmar Zander, Bojana V. Rosić, Alexander Litvinenko, Oliver Pajonk

Research output: Chapter in Book/Report/Conference proceedingChapter

19 Scopus citations

Abstract

In a Bayesian setting, inverse problems and uncertainty quantification (UQ)—the propagation of uncertainty through a computational (forward) model—are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. We give a detailed account of this approach via conditional approximation, various approximations, and the construction of filters. Together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update in form of a filter is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and nonlinear Bayesian update in form of a filter on some examples.
Original languageEnglish (US)
Title of host publicationComputational Methods for Solids and Fluids
PublisherSpringer Nature
Pages245-286
Number of pages42
ISBN (Print)978-3-319-27994-7
DOIs
StatePublished - Feb 13 2016

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