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Adaptive Bayesian Inference for Discontinuous Inverse Problems, Application to Hyperbolic Conservation Laws

Published online by Cambridge University Press:  03 June 2015

Alexandre Birolleau ,
Gaël Poëtte  and
Didier Lucor
Alexandre Birolleau*
Affiliation:
CEA, DAM, DIF, F-91297 Arpajon, France UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France
Gaël Poëtte*
Affiliation:
CEA, DAM, DIF, F-91297 Arpajon, France
Didier Lucor*
Affiliation:
UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France
*
Email:alexandre.biroleau@ea.fr
Email:gael.poette@cea.fr
Corresponding author.Email:didier.lucor@upmc.fr
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Abstract

Various works from the literature aimed at accelerating Bayesian inference in inverse problems. Stochastic spectral methods have been recently proposed as surrogate approximations of the forward uncertainty propagation model over the support of the prior distribution. These representations are efficient because they allow affordable simulation of a large number of samples from the posterior distribution. Unfortunately, they do not perform well when the forward model exhibits strong nonlinear behavior with respect to its input.

In this work, we first relate the fast (exponential) L2-convergence of the forward approximation to the fast (exponential) convergence (in terms of Kullback-Leibler divergence) of the approximate posterior. In particular, we prove that in case the prior distribution is uniform, the posterior is at least twice as fast as the convergence rate of the forwardmodel in those norms. The Bayesian inference strategy is developed in the framework of a stochastic spectral projectionmethod. The predicted convergence rates are then demonstrated for simple nonlinear inverse problems of varying smoothness.

We then propose an efficient numerical approach for the Bayesian solution of inverse problems presenting strongly nonlinear or discontinuous systemresponses. This comes with the improvement of the forward model that is adaptively approximated by an iterative generalized Polynomial Chaos-based representation. The numerical approximations and predicted convergence rates of the former approach are compared to the new iterative numericalmethod for nonlinear time-dependent test cases of varying dimension and complexity, which are relevant regarding our hydrodynamics motivations and therefore regarding hyperbolic conservation laws and the apparition of discontinuities in finite time.

Keywords

52B1065D1868U0568U07Stochastic inverse problemsBayesian inferenceiterative generalized Polynomial Chaoscompressible gas dynamics.

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 1 , July 2014 , pp. 1 - 34
Copyright
Copyright © Global Science Press Limited 2014

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