{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:24:24Z","timestamp":1787336664577,"version":"build-2736575974"},"reference-count":41,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1923221"],"award-info":[{"award-number":["DMS-1923221"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1929284"],"award-info":[{"award-number":["DMS-1929284"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,4,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We provide a unifying framework for [Formula: see text]-optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive the gradients of the [Formula: see text] cost function with respect to the reduced matrices, which then allows a nonintrusive, data-driven, gradient-based descent algorithm to construct the optimal approximant using only output samples. By choosing an appropriate measure, the framework covers both continuous (Lebesgue) and discrete cost functions. We show the efficacy of the proposed algorithm via various numerical examples. Furthermore, we analyze under what conditions the data-driven approximant can be obtained via projection.<\/jats:p>","DOI":"10.1137\/22m1500678","type":"journal-article","created":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T03:39:58Z","timestamp":1682480398000},"page":"A554-A578","source":"Crossref","is-referenced-by-count":14,"title":["\\(\\boldsymbol{\\mathcal{L}_2}\\)-Optimal Reduced-Order Modeling Using Parameter-Separable Forms"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9437-7698","authenticated-orcid":true,"given":"Petar","family":"Mlinari\u0107","sequence":"first","affiliation":[{"name":"Department of Mathematics, Virginia Tech, Blacksburg, VA 24061 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4564-5999","authenticated-orcid":true,"given":"Serkan","family":"Gugercin","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Division of Computational Modeling and Data Analytics, Academy of Data Science, Virginia Tech, Blacksburg, VA 24061 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,4,26]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718713"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-5757-3_1"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611976083"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2004.08.006"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/090776925"},{"key":"ref6","author":"Benner P.","year":"2019","journal-title":"Identification of Dominant Subspaces for Linear Structured Parametric Systems and Model Reduction"},{"key":"ref7","volume-title":"Model Order Reduction: Volume 2: Snapshot-Based Methods and Algorithms","author":"Benner P.","year":"2020"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1515\/9783110498967"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1137\/130932715"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974829"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1137\/15M1025426"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2018.11.032"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1137\/130916138"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1137\/140961511"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1137\/15M1010774"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1109\/TMTT.2019.2948858"},{"key":"ref17","unstructured":"A. R. Grimm , Parametric Dynamical Systems: Transient Analysis and Data Driven Modeling, Ph.D. thesis, Virginia Polytechnic Institute and State University, 2018, http:\/\/hdl.handle.net\/10919\/83840."},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1137\/060666123"},{"key":"ref19","unstructured":"S. Gugercin , \nA. C. Antoulas , and \nC. A. Beattie . A rational Krylov iteration for optimal \\(\\mathcal{H}_2\\) model reduction, in Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, 2006, pp. 1665\u20131667."},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1137\/130906635"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1109\/61.772353"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-020-2649-2"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-22470-1"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1155\/2014\/843869"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1137\/19M1247863"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1137\/21M140290X"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-27909-1_3"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1137\/15M1026614"},{"key":"ref29","unstructured":"T. Mitchell , Solving problems in reduced order modeling via nonsmooth constrained optimization and fast approximation, in Workshop on Model Reduction Methods and Optimization, 2016, https:\/\/www.mathos.unios.hr\/index.php\/443."},{"key":"ref30","unstructured":"P. Mlinari\u0107 , \\(\\mathcal{L}_2\\)-Optimal Reduced-Order Modeling Eexperiments, 2022, https:\/\/github.com\/pmli\/l2-opt-rom-ex\/tree\/v1."},{"key":"ref31","doi-asserted-by":"crossref","unstructured":"P. Mlinari\u0107  and \nS. Gugercin , A Unifying Framework for Interpolatory \\(\\mathcal{L}_2\\)-Optimal Reduced-Order Modeling, 2022, https:\/\/doi.org\/10.48550\/arXiv.2209.00714.","DOI":"10.1137\/22M1516920"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1137\/16M1106122"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1007\/b98874"},{"key":"ref34","unstructured":"Flexible Aircraft, Model Order Reduction Wiki, 2021, http:\/\/modelreduction.org\/index.php\/Flexible_Aircraft."},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2016.03.025"},{"key":"ref36","unstructured":"D. Petersson , A Nonlinear Optimization Approach to \\(\\mathcal{H}_{2}\\)-Optimal Modeling and Control, dissertation, Link\u00f6ping University, 2013, http:\/\/liu.diva-portal.org\/smash\/get\/diva2:647068\/FULLTEXT01.pdf."},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1016\/j.ifacol.2018.03.094"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-15431-2"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1963.1105517"},{"key":"ref40","unstructured":"P. Schwerdtner  and \nM. Voigt , SOBMOR: Structured Optimization-Based Model Order Reduction, https:\/\/arxiv.doi.org\/10.48550\/2011.07567, 2020."},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1038\/s41592-019-0686-2"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M1500678","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:25:46Z","timestamp":1787333146000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M1500678"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,26]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,4,30]]}},"alternative-id":["10.1137\/22M1500678"],"URL":"https:\/\/doi.org\/10.1137\/22m1500678","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,26]]}}}