Inverse Problems in a Bayesian Setting

H.G. Matthies, E. Zander, B.V. Rosić, A. Litvinenko, O. Pajonk
Computational Methods for Solids and Fluids Multiscale Analysis, Probability: Multiscale Analysis, Probability Aspects and Model Reduction, pp. 245-286, (2016)

Inverse Problems in a Bayesian Setting

Keywords

Inverse identification, Uncertainty quantification, Bayesian update, Parameter identification, Conditional expectation, Filters Functional, Spectral approximation

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.

Code

DOI: 10.1186/s40323-016-0075-7

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