Hierarchical off-diagonal low-rank approximation of Hessians in inverse problems, with application to ice sheet model initialization

Obtaining lightweight and accurate approximations of Hessian applies in inverse problems governed by partial differential equations (PDEs) is an essential task to make both deterministic and Bayesian statistical large-scale inverse problems computationally tractable. The $\mathcal{O}(N^{3})$ computational complexity of dense linear algebraic routines such as that needed for sampling from Gaussian proposal distributions and Newton solves by direct linear methods, can be reduced to log-linear complexity by utilizing hierarchical off-diagonal low-rank (HODLR) matrix approximations. In this work, we show that a class of Hessians that arise from inverse problems governed by PDEs are well approximated by the HODLR matrix format. In particular, we study inverse problems governed by PDEs that model the instantaneous viscous flow of ice sheets. In these problems, we seek a spatially distributed basal sliding parameter field such that the flow predicted by the ice sheet model is consistent with ice sheet surface velocity observations. We demonstrate the use of HODLR approximation by efficiently generating Hessian approximations that allow fast generation of samples from a Gaussianized posterior proposal distribution. Computational studies are performed which illustrate ice sheet problem regimes for which the Gauss-Newton data-misfit Hessian is more efficiently approximated by the HODLR matrix format than the low-rank (LR) format. We then demonstrate that HODLR approximations can be favorable, when compared to global low-rank approximations, for large-scale problems by studying the data-misfit Hessian associated to inverse problems governed by the Stokes flow model on the Humboldt glacier and Greenland ice sheets.