{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T21:09:51Z","timestamp":1790716191227,"version":"4.1.0"},"reference-count":34,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["GSC 111"],"award-info":[{"award-number":["GSC 111"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-FG02-08ER2585"],"award-info":[{"award-number":["DE-FG02-08ER2585"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0009297"],"award-info":[{"award-number":["DE-SC0009297"]}],"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":[[2017,1]]},"abstract":"<jats:p>Parameter optimization problems constrained by partial differential equations (PDEs) appear in many science and engineering applications. Solving these optimization problems may require a prohibitively large number of computationally expensive PDE solves, especially if the dimension of the design space is large. It is therefore advantageous to replace expensive high-dimensional PDE solvers (e.g., finite element) with lower-dimensional surrogate models. In this paper, the reduced basis (RB) model reduction method is used in conjunction with a trust region optimization framework to accelerate PDE-constrained parameter optimization. Novel a posteriori error bounds on the RB cost and cost gradient for quadratic cost functionals (e.g., least squares) are presented and used to guarantee convergence to the optimum of the high-fidelity model. The proposed certified RB trust region approach uses high-fidelity solves to update the RB model only if the approximation is no longer sufficiently accurate, reducing the number of full-fidelity solves required. We consider problems governed by elliptic and parabolic PDEs and present numerical results for a thermal fin model problem in which we are able to reduce the number of full solves necessary for the optimization by up to 86%.<\/jats:p>","DOI":"10.1137\/16m1081981","type":"journal-article","created":{"date-parts":[[2017,10,26]],"date-time":"2017-10-26T16:24:40Z","timestamp":1509035080000},"page":"S434-S460","source":"Crossref","is-referenced-by-count":46,"title":["A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization"],"prefix":"10.1137","volume":"39","author":[{"given":"Elizabeth","family":"Qian","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"Grepl","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Karen","family":"Veroy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Karen","family":"Willcox","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,10,26]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01197433"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2002.1184383"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/PL00011391"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623497325107"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.2514\/6.2008-5964"},{"key":"atypb6","unstructured":"R. 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