Abstract

Obtaining lightweight and accurate approximations of discretized objective functional Hessians in inverse problems governed by partial differential equations (PDEs) is essential to make both deterministic and Bayesian statistical large-scale inverse problems computationally tractable. The cubic computational complexity of dense linear algebraic tasks, such as Cholesky factorization, that provide a means to sample Gaussian distributions and determine solutions of Newton linear systems is a computational bottleneck at large-scale. These tasks 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. Here, we demonstrate the use of HODLR Hessian approximation to efficiently sample the Laplace approximation of the posterior distribution with covariance further approximated by HODLR matrix compression. Computational studies are performed which illustrate ice sheet problem regimes for which the Gauss–Newton data-misfit Hessianmore » 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 LR approximations, for large-scale problems by studying the data-misfit Hessian associated with inverse problems governed by the first-order Stokes flow model on the Humboldt glacier and Greenland ice sheet.« less

Authors:

Hartland, Tucker  ^[1];

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• Search DOE PAGES for ORCID "0000-0002-4638-3209"

Stadler, Georg  ^[2];

• Search DOE PAGES for author "Stadler, Georg"
• Search DOE PAGES for ORCID "0000-0001-7762-6544"

Perego, Mauro  ^[3];

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• Search DOE PAGES for ORCID "0000-0002-2671-8032"

Liegeois, Kim Anne J.  ^[3];

• Search DOE PAGES for author "Liegeois, Kim Anne J."
• Search DOE PAGES for ORCID "0000-0002-1182-4078"

Petra, Noémi  ^[1]

• Search DOE PAGES for author "Petra, Noémi"
• Search DOE PAGES for ORCID "0000-0002-9491-0034"

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+ Show Author Affiliations

1. Univ. of California, Merced, CA (United States)
2. New York Univ. (NYU), NY (United States)
3. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Publication Date:

Mon Jun 26 00:00:00 EDT 2023

Research Org.:

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Org.:

USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Biological and Environmental Research (BER); USDOE Office of Science (SC), Basic Energy Sciences (BES). Scientific User Facilities (SUF); USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)

OSTI Identifier:

2311406

Report Number(s):

SAND-2023-08485J
Journal ID: ISSN 0266-5611

Grant/Contract Number:  

NA0003525; AC02-05CH11231; DMS-1840265

Resource Type:

Accepted Manuscript

Journal Name:

Inverse Problems

Additional Journal Information:

Journal Volume: 39; Journal Issue: 8; Journal ID: ISSN 0266-5611

Publisher:

IOPscience

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; Hessians; inverse problems; hierarchical matrices; HODLR matrices; ice-sheet models; large-scale