Hessian-based Implicit Dimension Reduction for Large-scale Bayesian Inverse Problems
Omar Ghattas, Institute for Computational Engineering and Sciences, Jackson School of Geosciences, and Department of Mechanical Engineering
The University of Texas at Austin
Bayesian inference provides a systematic framework for quantifying uncertainty in the solution of ill-posed inverse problems. Given uncertainty in observational data, the forward model, and any prior knowledge of the parameters, the solution of the Bayesian inverse problem yields the so-called posterior probability of the model parameters, conditioned on the data. The fundamental challenge is how to explore this posterior density, in particular when the forward model is represented by PDEs and the uncertain parameters are given by (a discretized) infinite dimensional field. To overcome the prohibitive nature of Bayesian inversion for high-dimensional, expensive-to-evaluate models, we exploit the fact that, despite their large size, the observational data typically provide only sparse information on model parameters. This implicit dimension reduction is effected by low rank approximation of the data misfit Hessian, preconditioned by the prior covariance. We also discuss extensions of these ideas to low-rank approximations of higher-order derivative tensors.
The specific inverse problem we address here is the flow of ice from polar ice sheets such as Antarctica and Greenland, which is the primary contributor to projected sea level rise. One of the main difficulties faced in modeling ice sheet flow is the uncertain spatially-varying Robin boundary condition that describes the resistance to sliding at the base of the ice. Satellite observations of the surface ice flow velocity, along with a model of ice as a creeping incompressible shear-thinning fluid, can be used to infer this uncertain basal boundary condition. We cast this ill-posed inverse problem in the framework of Bayesian inference, which allows us to infer not only the basal sliding parameters, but also the associated uncertainty. We show results for Bayesian inversion of the basal sliding parameter field for the full Antarctic continent.
This work is joint with Nick Alger (UT-Austin), Tobin Isaac (UT-Austin), James Martin (UT-Austin), Noemi Petra (UC-Merced), and Georg Stadler (NYU).