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Inverse problems constrained by partial differential equations (PDEs) play acritical role in model development and calibration. In many applications, thereare multiple uncertain parameters in a model which must be estimated. Althoughthe Bayesian formulation is attractive for such problems, computational costand high dimensionality frequently prohibit a thorough exploration of theparametric uncertainty. A common approach is to reduce the dimension by fixingsome parameters (which we will call auxiliary parameters) to a best estimateand use techniques from PDE-constrained optimization to approximate propertiesof the Bayesian posterior distribution. For instance, the maximum a posterioriprobability (MAP) and the Laplace approximation of the posterior covariance canbe computed. In this article, we propose using hyper-differential sensitivityanalysis (HDSA) to assess the sensitivity of the MAP point to changes in theauxiliary parameters. We establish an interpretation of HDSA as correlations inthe posterior distribution. Our proposed framework is demonstrated on theinversion of bedrock topography for the Greenland ice sheet with uncertaintiesarising from the basal friction coefficient and climate forcing (iceaccumulation rate)

Hyper-differential sensitivity analysis in the context of Bayesian inference applied to ice-sheet problems