Solution of inverse problems with limited forward solver evaluations: a Bayesian perspective

Solving inverse problems based on computationally demanding forward models is ubiquitously difficult since one is necessarily limited to just a few observations of the response surface. The usual practice is to replace the response surface with a surrogate. However, this approach induces additional uncertainties on the posterior distributions. The main contribution of this work is the reformulation of the Bayesian solution of the inverse problem when the expensive forward model is replaced by the surrogate. We derive three approximations of the reformulated solution with increasing complexity and fidelity. We demonstrate numerically that the proposed approximations capture the uncertainty of the solution of the inverse problem induced by the fact that the forward model is replaced by a finite number of simulations. We demonstrate our approach in two different problems: locating the contamination source of a diffusive process and inferring the permeability field of an oil reservoir based on measurements of the oil-cut curves.