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) $L^2$-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 forward model in those norms. The Bayesian inference strategy is developed in the framework of a stochastic spectral projection method. 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 system responses. 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 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 of discontinuities in finite time.