Enabling Hyper-Differential Sensitivity Analysis for Ill-Posed Inverse Problems

Hart, Joseph Lee; van Bloemen Waanders, Bart G.

doi:10.1137/22m147699x

Enabling Hyper-Differential Sensitivity Analysis for Ill-Posed Inverse Problems

Journal Article · Wed Jul 26 00:00:00 EDT 2023 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/22m147699x· OSTI ID:2311559

Hart, Joseph Lee ^[1]; van Bloemen Waanders, Bart G. ^[1]

Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States). Scientific Machine Learning

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. Here 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 multiphysics inverse problem motivated by estimation of spatially heterogeneous material properties in the presence of spatially distributed parametric modeling uncertainties.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: NA0003525

OSTI ID:: 2311559

Report Number(s):: SAND2023-10309J

Journal Information:: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 4 Vol. 45; ISSN 1064-8275

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

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97 MATHEMATICS AND COMPUTING
PDE-constrained optimization
hyper-differential sensitivity analysis
inverse problems

Enabling Hyper-Differential Sensitivity Analysis for Ill-Posed Inverse Problems

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