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Iterative updating of model error for Bayesian inversion. (English) Zbl 1386.65056

The computational inverse problem is commonly described as a problem of the definition a forward map relating the unknown to an observed quantity, and to look for an estimate of the unknown when the data is corrupted by noise. Modelling the discrepancy between the reality and the noise is an active research topic in statistics.
In inverse problems common sources of modeling errors include:
(i)
model reduction – a complex, computationally intensive model is replaced by a simpler, loss demanding model;
(ii)
parametric reduction – in a model depending on poorly known parameters, some of them are frozen to fixed values, assuming that the solution is not sensitive to them;
(iii)
unknown geometry – a computational domain of unknown shape is approximated by a standard geometry.
The contributions of the topic above are the next:
(1)
The iterative updating of the posterior probability densities based on repeated updating of the model error distribution is developed, leading to an approximation of the posterior probability density;
(2)
The mean and covariances of the resulting sequence of posterior distributions converge to a limit geometrically fast when the models are linear and the noise and prior distributions are Gaussian;
(3)
The effectiveness of the algorithms are shown numerically in multiple different settings, showing the advantage over the conventional and enhanced error model.

MSC:

65C60 Computational problems in statistics (MSC2010)
62F15 Bayesian inference
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
92C55 Biomedical imaging and signal processing
78-05 Experimental work for problems pertaining to optics and electromagnetic theory
76-05 Experimental work for problems pertaining to fluid mechanics

Software:

DistMesh; Matlab
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Full Text: DOI arXiv Link

References:

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