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This paper introduces a statistical treatment of inverse problems constrainedby models with stochastic terms. The solution of the forward problem is givenby a distribution represented numerically by an ensemble of simulations. Thegoal is to formulate the inverse problem, in particular the objective function,to find the closest forward distribution (i.e., the output of the stochasticforward problem) that best explains the distribution of the observations in acertain metric. We use proper scoring rules, a concept employed in statisticalforecast verification, namely energy, variogram, and hybrid (i.e., combinationof the two) scores. We study the performance of the proposed formulation in thecontext of two applications: a coefficient field inversion for subsurface flowgoverned by an elliptic partial differential equation (PDE) with a stochasticsource and a parameter inversion for power grid governed bydifferential-algebraic equations (DAEs). In both cases we show that thevariogram and the hybrid scores show better parameter inversion results thandoes the energy score, whereas the energy score leads to better probabilisticpredictions.

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