Sensitivity Analysis of the Information Gain in Infinite-Dimensional Bayesian Linear Inverse Problems

IF 1.5 4区工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY

International Journal for Uncertainty Quantification Pub Date : 2024-05-01 DOI:10.1615/int.j.uncertaintyquantification.2024051416

Abhijit Chowdhary, Shanyin Tong, Georg Stadler, Alen Alexanderian

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Abstract

We study the sensitivity of infinite-dimensional Bayesian linear inverse problems governed by partial differential equations (PDEs) with respect to modeling uncertainties. In particular, we consider derivative-based sensitivity analysis of the information gain, as measured by the Kullback-Leibler divergence from the posterior to the prior distribution. To facilitate this, we develop a fast and accurate method for computing derivatives of the information gain with respect to auxiliary model parameters. Our approach combines low-rank approximations, adjoint-based eigenvalue sensitivity analysis, and post-optimal sensitivity analysis. The proposed approach also paves way for global sensitivity analysis by computing derivative-based global sensitivity measures. We illustrate different aspects of the proposed approach using an inverse problem governed by a scalar linear elliptic PDE, and an inverse problem governed by the three-dimensional equations of linear elasticity, which is motivated by the inversion of the fault-slip field after an earthquake.

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无穷维贝叶斯线性逆问题中信息增益的敏感性分析

我们研究了由偏微分方程（PDE）控制的无限维贝叶斯线性逆问题对建模不确定性的敏感性。特别是，我们考虑对信息增益进行基于导数的敏感性分析，信息增益是通过后验分布与先验分布之间的库尔贝克-莱布勒发散来衡量的。为了便于分析，我们开发了一种快速准确的方法，用于计算信息增益与辅助模型参数的导数关系。我们的方法结合了低阶近似、基于邻接的特征值灵敏度分析和后优化灵敏度分析。所提出的方法还通过计算基于导数的全局灵敏度度量，为全局灵敏度分析铺平了道路。我们用一个标量线性椭圆 PDE 所控制的逆问题和一个线性弹性三维方程所控制的逆问题来说明所提方法的不同方面。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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.