We consider hyper-differential sensitivity analysis (HDSA) of nonlinear Bayesian inverse problems governed by partial differential equations (PDEs) with infinite-dimensional parameters. In previous works, HDSA has been used to assess the sensitivity of the solution of deterministic inverse problems to additional model uncertainties and also different types of measurement data. In the present work, we extend HDSA to the class of Bayesian inverse problems governed by PDEs. The focus is on assessing the sensitivity of certain key quantities derived from the posterior distribution. Specifically, we focus on analyzing the sensitivity of the MAP point and the Bayes risk and make full use of the information embedded in the Bayesian inverse problem. After establishing our mathematical framework for HDSA of Bayesian inverse problems, we present a detailed computational approach for computing the proposed HDSA indices. We examine the effectiveness of the proposed approach on an inverse problem governed by a PDE modeling heat conduction.

中文翻译：

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非线性贝叶斯反问题的超微分敏感性分析

我们考虑由具有无限维参数的偏微分方程 (PDE) 控制的非线性贝叶斯反问题的超微分灵敏度分析 (HDSA)。在之前的工作中，HDSA 已用于评估确定性反问题的解决方案对其他模型不确定性以及不同类型的测量数据的敏感性。在目前的工作中，我们将 HDSA 扩展到由偏微分方程控制的贝叶斯逆问题类。重点是评估从后验分布得出的某些关键量的敏感性。具体来说，我们重点分析MAP点的敏感性和贝叶斯风险，并充分利用贝叶斯逆问题中嵌入的信息。在建立了贝叶斯反问题 HDSA 的数学框架后，我们提出了一种详细的计算方法来计算所提出的 HDSA 指数。我们检查了所提出的方法在由偏微分方程热传导建模控制的反问题上的有效性。