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


Abstract: We consider sensitivity analysis of Bayesian linear inverse problems with respect to modeling uncertainties. To this end, we consider sensitivity analysis of the information gain, as measured by the Kullback—Leibler divergence from the posterior to the prior. This choice provides a principled approach that leverages key structures within the Bayesian inverse problem. Also, the information gain admits a closed-form expression in the case of linear Gaussian inverse problems. The derivatives of the information gain are extremely challenging to compute. To address this challenge, we present accurate and efficient methods that combine eigenvalue sensitivities and hyper-differential sensitivity analysis that take advantage of adjoint based gradient and Hessian computation. This results in a computational approach whose cost, in number of PDE solves, does not grow upon mesh refinement. These results are presented in an application-driven model problem, considering a simplified earthquake model to infer fault slip from surface measurements.