Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model which must be estimated. Although the Bayesian formulation is attractive for such problems, computational cost and high dimensionality frequently prohibit a thorough exploration of the parametric uncertainty. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and use techniques from PDE-constrained optimization to approximate properties of the Bayesian posterior distribution. For instance, the maximum a posteriori probability (MAP) and the Laplace approximation of the posterior covariance can be computed. In this article, we propose using hyperdifferential sensitivity analysis (HDSA) to assess the sensitivity of the MAP point to changes in the auxiliary parameters. We establish an interpretation of HDSA as correlations in the posterior distribution. Our proposed framework is demonstrated on the inversion of bedrock topography for the Greenland ice-sheet with uncertainties arising from the basal friction coefficient and climate forcing (ice accumulation rate).

中文翻译：

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应用于冰原问题的贝叶斯推理背景下的超微分敏感性分析


受偏微分方程 (PDE) 约束的反问题在模型开发和校准中发挥着关键作用。在许多应用中，模型中有多个必须估计的不确定参数。尽管贝叶斯公式对于此类问题很有吸引力，但计算成本和高维度经常阻碍对参数不确定性的彻底探索。一种常见的方法是通过固定一些参数（我们称之为辅助参数）来减少维数以达到最佳估计，并使用 PDE 约束优化技术来近似贝叶斯后验分布的属性。例如，可以计算后验协方差的最大后验概率（MAP）和拉普拉斯近似。在本文中，我们建议使用超微分敏感性分析（HDSA）来评估 MAP 点对辅助参数变化的敏感性。我们将 HDSA 解释为后验分布中的相关性。我们提出的框架通过格陵兰冰盖基岩地形的反演进行了论证，其不确定性来自于基础摩擦系数和气候强迫（冰积累率）。