Statistical Treatment of Inverse Problems Constrained by Differential Equations-Based Models with Stochastic Terms
Statistical Treatment of Inverse Problems Constrained by Differential Equations-Based Models with Stochastic Terms
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DOI：
10.1137/18m122073x
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发表时间：
2018-10
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影响因子：
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通讯作者：
E. Constantinescu;N. Petra;J. Bessac;C. Petra
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文献类型：
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作者：
E. Constantinescu;N. Petra;J. Bessac;C. Petra
本文介绍了一种带随机项的模型约束反问题的统计处理方法。前向问题的解决方案是由一个分布表示数值模拟合奏。目标是用公式表示逆问题，特别是目标函数，以找到最接近的正向分布（即，随机前向问题的输出），其最好地解释了在特定度量中的观测的分布。我们使用适当的评分规则，这是统计预测验证中采用的一个概念，即能量、变异函数和混合（即，两者的组合）得分。我们研究了两个应用程序的背景下，所提出的配方的性能：一个系数场反演地下流由椭圆偏微分方程（PDE）与随机源和参数反演由微分代数方程（DAE）的电网。在这两种情况下，我们表明，变差函数和混合分数比能量分数显示更好的参数反演结果，而能量分数导致更好的概率预测。
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.