论文标题
非线性贝叶斯逆问题的超差异敏感性分析
Hyper-differential sensitivity analysis for nonlinear Bayesian inverse problems
论文作者
论文摘要
我们考虑了由具有无限维参数的PDE的非线性贝叶斯逆问题的超差异敏感性分析（HDSA）。在以前的工作中，HDSA已用于评估确定性逆问题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PDE管辖的贝叶斯反问题类。重点是评估从后验分布得出的某些关键量的灵敏度。具体而言，我们专注于分析地图点和贝叶斯风险的敏感性，并充分利用贝叶斯反问题中嵌入的信息。在建立了针对贝叶斯逆问题HDSA的数学框架之后，我们提出了一种计算拟议HDSA指数的详细计算方法。我们检查了拟议方法对由PDE进行热传导控制的模型反问题的有效性。
We consider hyper-differential sensitivity analysis (HDSA) of nonlinear Bayesian inverse problems governed by 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 a model inverse problem governed by a PDE for heat conduction.
