{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:40:44Z","timestamp":1787341244921,"version":"build-2736575974"},"reference-count":46,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM\/ASA J. Uncertainty Quantification"],"published-print":{"date-parts":[[2023,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>The ensemble Kalman inversion (EKI) for the solution of Bayesian inverse problems of type [Formula: see text], with [Formula: see text] being an unknown parameter, [Formula: see text] a given datum, and [Formula: see text] measurement noise, is a powerful tool usually derived from a sequential Monte Carlo point of view. It describes the dynamics of an ensemble of particles [Formula: see text], whose initial empirical measure is sampled from the prior, evolving over an artificial time [Formula: see text] toward an approximate solution of the inverse problem, with [Formula: see text] emulating the posterior, and [Formula: see text] corresponding to the underregularized minimum-norm solution of the inverse problem. Using spectral techniques, we provide a complete description of the deterministic dynamics of EKI and its asymptotic behavior in parameter space. In particular, we analyze the dynamics of naive EKI and mean-field EKI with a special focus on their time asymptotic behavior. Furthermore, we show that\u2014even in the deterministic case\u2014residuals in parameter space do not decrease monotonously in the Euclidean norm and suggest a problem-adapted norm, where monotonicity can be proved. Finally, we derive a system of ordinary differential equations governing the spectrum and eigenvectors of the covariance matrix. While the analysis is aimed at the EKI, we believe that it can be applied to understand more general particle-based dynamical systems.<\/jats:p>","DOI":"10.1137\/21m1429461","type":"journal-article","created":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T11:59:45Z","timestamp":1678708785000},"page":"320-357","source":"Crossref","is-referenced-by-count":4,"title":["Complete Deterministic Dynamics and Spectral Decomposition of the Linear Ensemble Kalman Inversion"],"prefix":"10.1137","volume":"11","author":[{"given":"Leon","family":"Bungert","sequence":"first","affiliation":[{"name":"Hausdorff Center for Mathematics, University of Bonn, Villa Maria, 53115 Bonn, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Philipp","family":"Wacker","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,3,11]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1002\/qj.2186"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1137\/21M1414000"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/140964023"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1002\/qj.457"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1002\/qj.672"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1002\/qj.672"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1127\/0941-2948\/2012\/0307"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1175\/1520-0493(2001)129<0420:ASWTET>2.0.CO;2"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/ab149c"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1137\/17M1132367"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/ab1d71"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s00028-019-00545-1"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.2140\/apde.2021.14.823"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1137\/15M1054687"},{"key":"ref15","unstructured":"E. 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