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：我们研究了由偏微分方程（PDEs）控制的无维贝叶斯线性反问题相对于模型不确定性的灵敏度。特别地，我们考虑了基于导数的信息增益灵敏度分析，该信息增益通过后验分布与先验分布之间的Kullback-Leibler散度来衡量。为实现此目的，我们开发了一种快速且精确的方法，用于计算信息增益关于辅助模型参数的导数。我们的方法结合了低秩近似、基于伴随的特征值灵敏度分析以及后最优灵敏度分析。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。我们通过一个由标量线性椭圆型PDE控制的反问题，以及一个由三维线性弹性方程控制的反问题（该问题源于地震后断层滑动场的反演），展示了所提出方法的不同方面。