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

Various works from the literature aimed at accelerating Bayesian inferencein inverse problems. Stochastic spectral methods have been recently proposed as surrogate approximations of the forward uncertainty propagation model over the supportof 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 nonlinearbehavior with respect to its input.In this work, we first relate the fast (exponential) $L^2$-convergence of the forwardapproximation to the fast (exponential) convergence (in terms of Kullback-Leibler divergence) of the approximate posterior. In particular, we prove that in case the priordistribution is uniform, the posterior is at least twice as fast as the convergence rate ofthe forward model in those norms. The Bayesian inference strategy is developed in theframework of a stochastic spectral projection method. The predicted convergence ratesare 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 system responses. Thiscomes with the improvement of the forward model that is adaptively approximated byan iterative generalized Polynomial Chaos-based representation. The numerical approximations and predicted convergence rates of the former approach are comparedto the new iterative numerical method 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 ofdiscontinuities in finite time.