  Inverse problems constrained by partial differential equations (PDEs) play a
critical role in model development and calibration. In many applications, there
are multiple uncertain parameters in a model that must be estimated. However,
high dimensionality of the parameters and computational complexity of the PDE
solves make such problems challenging. A common approach is to reduce the
dimension by fixing some parameters (which we will call auxiliary parameters)
to a best estimate and use techniques from PDE-constrained optimization to
estimate the other parameters. In this article, hyper-differential sensitivity
analysis (HDSA) is used to assess the sensitivity of the solution of the
PDE-constrained optimization problem to changes in the auxiliary parameters.
Foundational assumptions for HDSA require satisfaction of the optimality
conditions which are not always practically feasible as a result of
ill-posedness in the inverse problem. We introduce novel theoretical and
computational approaches to justify and enable HDSA for ill-posed inverse
problems by projecting the sensitivities on likelihood informed subspaces and
defining a posteriori updates. Our proposed framework is demonstrated on a
nonlinear multi-physics inverse problem motivated by estimation of spatially
heterogenous material properties in the presence of spatially distributed
parametric modeling uncertainties.
