{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:38:40Z","timestamp":1787330320431,"version":"build-2736575974"},"reference-count":34,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100006168","name":"National Nuclear Security Administration","doi-asserted-by":"publisher","award":["DE-NA-0003525"],"award-info":[{"award-number":["DE-NA-0003525"]}],"id":[{"id":"10.13039\/100006168","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006168","name":"National Nuclear Security Administration","doi-asserted-by":"publisher","award":["SAND2021-6604 O"],"award-info":[{"award-number":["SAND2021-6604 O"]}],"id":[{"id":"10.13039\/100006168","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["20-023231"],"award-info":[{"award-number":["20-023231"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,8,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>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. 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.<\/jats:p>","DOI":"10.1137\/22m147699x","type":"journal-article","created":{"date-parts":[[2023,7,26]],"date-time":"2023-07-26T04:45:52Z","timestamp":1690346752000},"page":"A1947-A1970","source":"Crossref","is-referenced-by-count":3,"title":["Enabling Hyper-Differential Sensitivity Analysis for Ill-Posed Inverse Problems"],"prefix":"10.1137","volume":"45","author":[{"given":"Joseph","family":"Hart","sequence":"first","affiliation":[{"name":"Scientific Machine Learning, Sandia National Laboratories, Albuquerque, NM 87123-1320 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bart","family":"van Bloemen Waanders","sequence":"additional","affiliation":[{"name":"Scientific Machine Learning, Sandia National Laboratories, Albuquerque, NM 87123-1320 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,7,26]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/19M1270367"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4939-8636-1"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/28\/11\/115016"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/17\/3\/314"},{"key":"ref5","volume-title":"Large-Scale Inverse Problems and Quantification of Uncertainty","author":"Biegler L.","year":"2011"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718935"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-55508-4"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1137\/S106482750241565X"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827502415661"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144596302644"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2005.12.023"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1137\/12089586X"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/30\/11\/114015"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1081\/NFA-120034120"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/17\/6\/319"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1137\/090771806"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1615\/Int.J.UncertaintyQuantification.2020032480"},{"key":"ref18","unstructured":"J. Hart , \nB. van Bloemen Waanders , \nL. Hood , and \nJ. Parish , Sensitivity driven experimental design to facilitate control of dynamical systems, https:\/\/arxiv.org\/abs\/2202.03312, 2022."},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1137\/05062442X"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1137\/S105262340342907X"},{"key":"ref21","volume-title":"Optimization with PDE Constraints","author":"Hinze M.","year":"2009"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.04.047"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1007\/b138659"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-5981-1"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9496(2005)131:2(125)"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/aad91e"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1137\/130934805"},{"key":"ref28","doi-asserted-by":"crossref","first-page":"e2364","DOI":"10.1002\/nla.2364","author":"Saibaba A. K.","year":"2021","journal-title":"Numer. Linear Algebra Appl."},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2026"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492910000061"},{"key":"ref31","unstructured":"I. Sunseri , \nA. Alexanderian , \nJ. Hart , and \nB. van Bloemen Waanders , Hyper-differential Sensitivity Analysis for Nonlinear Bayesian Inverse Problems Governed by PDEs, 2022, https:\/\/arxiv.org\/abs\/2202.02219."},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/abaf63"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897317703"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717570"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M147699X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:57:08Z","timestamp":1787327828000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M147699X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,26]]},"references-count":34,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,8,31]]}},"alternative-id":["10.1137\/22M147699X"],"URL":"https:\/\/doi.org\/10.1137\/22m147699x","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,26]]}}}