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

TL;DR

The paper tackles the sensitivity of posterior information gain to auxiliary PDE parameters in infinite-dimensional Bayesian linear inverse problems. It introduces a scalable framework that combines low-rank approximations of the prior-preconditioned data-misfit Hessian, adjoint-based eigenvalue sensitivity analysis, and post-optimal sensitivity analysis to compute derivatives of the information gain and its expectation with respect to ${\boldsymbol{\theta}}$. It also provides derivative-based global sensitivity bounds for Sobol indices and demonstrates the approach on elliptic source inversion and 3D fault-slip inversion in elasticity, revealing that certain Lamé parameters dominate sensitivity while others have limited impact. The work offers practical guidance for model specification and parameter prioritization, and lays a foundation for extending to nonlinear problems and Bayesian optimal experimental design under uncertainty.

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.

Figures (7)

• Figure 1: Information gain sensitivities for two-dimensional model problem. Shown are the absolute values of the derivatives with respect to the auxiliary parameters.
• Figure 2: $\Phi_{\rm IG}$ and $\overline{\Phi}_{\rm IG}$ as functions of $g$ ($x$-axis) and $c$ ($y$-axis). Note the difference in scales and that $\overline{\Phi}_{\rm IG}$ is independent of $g$.
• Figure 3: (Left) Visualization of $\Omega$. (Right) Realization of $\lambda$ with six KLE modes.
• Figure 4: True slip field ${\hbox{\boldmath${m}$}}$ given by \ref{['eq:true-slip']} (left) and the slip MAP estimate ${\hbox{\boldmath${m}$}}$ (right).
• Figure 5: Fault slip (red arrows) and corresponding deformation on the top surface $\Gamma_t$. Shown on the left is the result for the truth fault slip used to generate synthetic data, and on the right the MAP fault slip.