Large-scale Bayesian inversion of the basal friction coefficient for the Antarctic ice sheet

Friday, 19 December 2014: 1:40 PM
Noemi Petra1, Tobin Isaac2, Georg Stadler3 and Omar Ghattas2,4, (1)University of California Merced, School of Natural Sciences, Merced, CA, United States, (2)The University of Texas at Austin, Institute for Computational Engineering & Sciences, Austin, TX, United States, (3)New York University, Courant Institute of Mathematical Sciences, New York, NY, United States, (4)The University of Texas at Austin, Jackson School of Geosciences, Austin, TX, United States
Model-based projections of the dynamics of the polar ice sheets play a
central role in anticipating future sea level rise. However, a number
of mathematical and computational challenges place significant
barriers on improving predictability of these models. One such
challenge is caused by the unknown model parameters that must be
inferred from heterogeneous observational data, leading to an
ill-posed inverse problem and the need to quantify uncertainties in
its solution. In this talk we discuss the problem of estimating the
uncertainty in the solution of (large-scale) ice sheet inverse
problems within the framework of Bayesian inference. The ice flow is
modeled as a three-dimensional, creeping, viscous, incompressible,
non-Newtonian fluid via the nonlinear Stokes equations.

Solving Bayesian inverse problems with expensive forward models and
high-dimensional parameter spaces is intractable on current and
anticipated supercomputers. However, under the assumption of Gaussian
noise and prior probability densities, and after linearizing the
parameter-to-observable map, the posterior density becomes Gaussian,
and can therefore be characterized by its mean and covariance. The
mean is given by the solution of a large-scale PDE-constrained
optimization problem and the posterior covariance matrix is given by
the inverse of the Hessian of the regularized data misfit
functional. Direct computation of the Hessian matrix is prohibitive,
since it would require solution of as many forward Stokes problems as
there are parameters. Therefore, we exploit the compact nature of the
data misfit component of Hessian and construct its low rank
approximation, which can be constructed at a cost (measured in number
of Stokes solves) that does not depend on the parameter or data
dimensions, thus providing scalability to problem sizes of practical
interest. We apply this framework to quantify uncertainties in the
inference of the basal friction coefficient for the Antarctic ice
sheet from InSAR satellite data.