Statistical Treatment of Inverse Problems Constrained by Differential Equations-Based Models with Stochastic Terms

This paper introduces a statistical treatment of inverse problems constrained by models with stochastic terms. The solution of the forward problem is given by a distribution represented numerically by an ensemble of simulations. The goal is to formulate the inverse problem, in particular the objective function, to find the closest forward distribution (i.e., the output of the stochastic forward problem) that best explains the distribution of the observations in a certain metric. We use proper scoring rules, a concept employed in statistical forecast verification, namely energy, variogram, and hybrid (i.e., combination of the two) scores. We study the performance of the proposed formulation in the context of two applications: a coefficient field inversion for subsurface flow governed by an elliptic partial differential equation (PDE) with a stochastic source and a parameter inversion for power grid governed by differential-algebraic equations (DAEs). In both cases we show that the variogram and the hybrid scores show better parameter inversion results than does the energy score, whereas the energy score leads to better probabilistic predictions.