SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 1, Page 170-197, January 2020.
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 with a stochastic source and a parameter inversion for power grid governed by differential-algebraic equations. In both cases we show that the variogram and the hybrid scores produce better parameter inversion results than does the energy score, whereas the energy score leads to better probabilistic predictions.

中文翻译：

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基于带有随机项的基于微分方程的模型约束的逆问题的统计处理

SIAM / ASA不确定性量化杂志，第8卷，第1期，第170-197页，2020年1月。
本文介绍了具有随机项的模型约束的逆问题的统计处理。前向问题的解决方案由一组仿真数值表示的分布给出。目的是制定反问题，特别是目标函数，以找到最能解释特定度量中观测值分布的最接近的正向分布（即，随机正向问题的输出）。我们使用适当的评分规则，这是统计预测验证中使用的概念，即能量，变异函数和混合（即，两者的结合）得分。我们在两种应用的背景下研究了拟议配方的性能：由具有随机源的椭圆偏微分方程控制的地下流的系数场反演和由微分代数方程控制的电网的参数反演。在这两种情况下，我们都表明，变异函数图和混合分数比能量分数产生更好的参数反演结果，而能量分数带来更好的概率预测。