  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.
