Hyper-differential sensitivity analysis for nonlinear Bayesian inverse problems

IF 1.5 4区工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY International Journal for Uncertainty Quantification Pub Date : 2023-01-01 DOI:10.1615/int.j.uncertaintyquantification.2023045300

Isaac Sunseri, Alen Alexanderian, Joseph Hart, Bart Van Bloemen Waanders

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Abstract

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.

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非线性贝叶斯反问题的超微分灵敏度分析

研究了具有无限维参数的偏微分方程控制的非线性贝叶斯反问题的超微分灵敏度分析(HDSA)。在以前的工作中，HDSA已被用于评估确定性逆问题解对附加模型不确定性和不同类型测量数据的敏感性。在本工作中，我们将HDSA推广到一类由偏微分方程控制的贝叶斯逆问题。重点是评估从后验分布得出的某些关键数量的敏感性。具体来说，我们着重分析了MAP点的敏感性和贝叶斯风险，充分利用了贝叶斯反问题所包含的信息。在建立了贝叶斯反问题HDSA的数学框架后，我们给出了计算所提出的HDSA指标的详细计算方法。我们检验了所提出的方法在由PDE控制的热传导模型逆问题上的有效性。

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来源期刊

International Journal for Uncertainty Quantification ENGINEERING, MULTIDISCIPLINARY-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

3.60

自引率

5.90%

发文量

期刊介绍： The International Journal for Uncertainty Quantification disseminates information of permanent interest in the areas of analysis, modeling, design and control of complex systems in the presence of uncertainty. The journal seeks to emphasize methods that cross stochastic analysis, statistical modeling and scientific computing. Systems of interest are governed by differential equations possibly with multiscale features. Topics of particular interest include representation of uncertainty, propagation of uncertainty across scales, resolving the curse of dimensionality, long-time integration for stochastic PDEs, data-driven approaches for constructing stochastic models, validation, verification and uncertainty quantification for predictive computational science, and visualization of uncertainty in high-dimensional spaces. Bayesian computation and machine learning techniques are also of interest for example in the context of stochastic multiscale systems, for model selection/classification, and decision making. Reports addressing the dynamic coupling of modern experiments and modeling approaches towards predictive science are particularly encouraged. Applications of uncertainty quantification in all areas of physical and biological sciences are appropriate.