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.

翻译：我们研究由偏微分方程（PDEs）支配的无限维贝叶斯线性反问题对建模不确定性的敏感性。具体而言，我们考虑基于导数的信息增益敏感性分析，其中信息增益由后验分布相对于先验分布的Kullback-Leibler散度度量。为此，我们开发了一种快速且精确的方法来计算信息增益关于辅助模型参数的导数。该方法结合了低秩近似、基于伴随矩阵的特征值敏感性分析以及最优后敏感性分析。提出的方法还通过计算基于导数的全局敏感性度量，为全局敏感性分析铺平了道路。我们利用一个由标量线性椭圆型PDE支配的反问题，以及一个由三维线性弹性方程支配的反问题（后者源于地震后断层滑动场反演的实际需求），展示了所提方法的不同方面。