{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:31:03Z","timestamp":1787232663367,"version":"build-2736575974"},"reference-count":79,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-24-1-0105"],"award-info":[{"award-number":["FA9550-24-1-0105"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006132","name":"Office of Science","doi-asserted-by":"publisher","award":["DE-SC0024721"],"award-info":[{"award-number":["DE-SC0024721"]}],"id":[{"id":"10.13039\/100006132","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2318880"],"award-info":[{"award-number":["DMS-2318880"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[2025,12,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Ensemble Kalman inversion (EKI) methods are a family of iterative methods for solving weighted least squares problems, especially those arising in scientific and engineering inverse problems in which unknown parameters or states are estimated from observed data by minimizing the weighted square norm of the data misfit. Implementation of EKI only requires the\u00a0evaluation of the forward model mapping the unknown to the data, and does not require derivatives or adjoints of the forward model. The methods therefore offer an attractive alternative to gradient-based optimization approaches in inverse problem settings where evaluating derivatives or adjoints of the forward model is computationally intractable. This work presents a new analysis of the behavior of both deterministic and stochastic versions of basic EKI for linear observation operators, resulting in a natural interpretation of EKI\u2019s convergence properties in terms of \u201cfundamental subspaces\u201d analogous to Strang\u2019s fundamental subspaces of linear algebra. Our analysis directly examines the discrete EKI iterations instead of their continuous-time limits considered in previous analyses, and it provides spectral decompositions that define six fundamental subspaces of EKI spanning both observation and state spaces. This approach verifies convergence rates previously derived for continuous-time limits, and yields new results describing both deterministic and stochastic EKI convergence behavior with respect to the standard minimum-norm weighted least squares solution in terms of the fundamental subspaces. Numerical experiments illustrate our theoretical results.<\/jats:p>","DOI":"10.1137\/24m1693143","type":"journal-article","created":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T08:28:09Z","timestamp":1762417689000},"page":"771-798","source":"Crossref","is-referenced-by-count":0,"title":["The Fundamental Subspaces of Ensemble Kalman Inversion"],"prefix":"10.1137","volume":"67","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6713-3746","authenticated-orcid":true,"given":"Elizabeth","family":"Qian","sequence":"first","affiliation":[{"name":"School of Aerospace Engineering, School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3302-4845","authenticated-orcid":true,"given":"Christopher","family":"Beattie","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24060 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,11,6]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1109\/18.985947"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/ab1c09"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/21M1414000"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1093\/gji\/ggac201"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1002\/qj.591"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1002\/qj.672"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611977950"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1137\/17M1132367"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/ab149c"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1137\/21M1437561"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.3934\/fods.2020004"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1137\/21M1429461"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492924000060"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1142\/9789811266140_0004"},{"key":"ref15","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1090\/mcom\/3709","volume":"91","author":"Chada N.","year":"2022","journal-title":"Math. 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A.","year":"2024","journal-title":"Inf. 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J. Palanthandalam-Madapusi, A. Girard, and D. S. 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Math. Appl."},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111559"},{"key":"ref65","doi-asserted-by":"publisher","DOI":"10.1029\/2021MS002937"},{"key":"ref66","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1993.11990500"},{"key":"ref67","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/accb08"},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.1093\/gji\/ggae012"},{"key":"ref69","volume-title":"Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements","author":"Twomey S.","year":"2013"},{"key":"ref70","doi-asserted-by":"publisher","DOI":"10.1002\/qj.3819"},{"key":"ref71","doi-asserted-by":"publisher","DOI":"10.1080\/16000870.2018.1445364"},{"key":"ref72","doi-asserted-by":"publisher","DOI":"10.1137\/21M1404119"},{"key":"ref73","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/ac8bed"},{"key":"ref74","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/ac5729"},{"key":"ref75","first-page":"iaae030","volume":"13","author":"Weissmann S.","year":"2024","journal-title":"Inf. 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