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.
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This work was partially supported by the German research foundation “Deutsche Forschungsgemeinschaft” (DFG).
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Matthies, H.G., Zander, E., Rosić, B.V., Litvinenko, A., Pajonk, O. (2016). Inverse Problems in a Bayesian Setting. In: Ibrahimbegovic, A. (eds) Computational Methods for Solids and Fluids. Computational Methods in Applied Sciences, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-319-27996-1_10
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DOI: https://doi.org/10.1007/978-3-319-27996-1_10
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