{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:44:29Z","timestamp":1787341469825,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["FA9550-17-1-0195"],"award-info":[{"award-number":["FA9550-17-1-0195"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000143","name":"Division of Computing and Communication Foundations","doi-asserted-by":"publisher","award":["CCF-1816219"],"award-info":[{"award-number":["CCF-1816219"]}],"id":[{"id":"10.13039\/100000143","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["DMS-1522798"],"award-info":[{"award-number":["DMS-1522798"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["DMS-1819144"],"award-info":[{"award-number":["DMS-1819144"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM\/ASA J. Uncertainty Quantification"],"published-print":{"date-parts":[[2020,1]]},"abstract":"<jats:p>This paper shows how to systematically and efficiently improve a reduced-order model (ROM) to obtain a better ROM-based estimate of the Conditional Value-at-Risk (CVaR) of a computationally expensive quantity of interest (QoI). Efficiency is gained by exploiting the structure of CVaR, which implies that a ROM used for CVaR estimation only needs to be accurate in a small region of the parameter space, called the $\\epsilon$-risk region. Hence, any full-order model (FOM) queries needed to improve the ROM can be restricted to this small region of the parameter space, thereby substantially reducing the computational cost of ROM construction. However, an example is presented which shows that simply constructing a new ROM that has a smaller error with the FOM is in general not sufficient to yield a better CVaR estimate. Instead a combination of previous ROMs is proposed that achieves a guaranteed improvement, as well as $\\epsilon$-risk regions that converge monotonically to the FOM risk region with decreasing ROM error. Error estimates for the ROM-based CVaR estimates are presented. The gains in efficiency obtained by improving a ROM only in the small $\\epsilon$-risk region over a traditional greedy procedure on the entire parameter space are illustrated numerically.<\/jats:p>","DOI":"10.1137\/19m1257433","type":"journal-article","created":{"date-parts":[[2020,5,5]],"date-time":"2020-05-05T14:25:30Z","timestamp":1588688730000},"page":"668-692","source":"Crossref","is-referenced-by-count":16,"title":["Adaptive Reduced-Order Model Construction for Conditional Value-at-Risk Estimation"],"prefix":"10.1137","volume":"8","author":[{"given":"Matthias","family":"Heinkenschloss","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Boris","family":"Kramer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Timur","family":"Takhtaganov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,5,5]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2013.08.016"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"B. Haasdonk,\n                      Chapter 2: Reduced basis methods for parametrized PDEs-a tutorial introduction for stationary and instationary problems\n                      , in Model Reduction and Approximation: Theory and Algorithms, P. Benner, A. Cohen, M. Ohlberger, and K. Willcox, eds., Computational Science and Engineering 15, SIAM, Philadelphia, 2017, pp. 65-136,https:\/\/doi.org\/10.1137\/1.9781611974829.ch2.","DOI":"10.1137\/1.9781611974829.ch2"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/17M1160069"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"J. S. Hesthaven, G. Rozza, and B. Stamm,\n                      Certified Reduced Basis Methods for Parametrized Partial Differential Equations\n                      , Springer Briefs in Mathematics, Springer, New York, 2015,https:\/\/doi.org\/10.1007\/978-3-319-22470-1.","DOI":"10.1007\/978-3-319-22470-1"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1145\/2661631"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/140955665"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/140954556"},{"key":"atypb8","first-page":"11","volume":"4","author":"Krokhmal P.","year":"2002","journal-title":"J. Risk"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(00)00275-9"},{"key":"atypb10","unstructured":"L. Machiels, Y. Maday, A. T. Patera, and D. V. Rovas,\n                      A blackbox reduced-basis output bound method for shape optimization\n                      , in Proceedings of the 12th International Conference on Domain Decomposition Methods, Chiba, Japan, T. Chan, T. Kako, and H. K. O. Pironneau, eds., DDM.org, 2001, pp. 429-436."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s10479-006-0142-4"},{"key":"atypb12","first-page":"1","author":"Norton M.","year":"2019","journal-title":"Ann. Oper. Res."},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/16M1082469"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1115\/1.1448332"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"A. Quarteroni, A. Manzoni, and F. Negri,\n                      Reduced Basis Methods for Partial Differential Equations. An Introduction\n                      , Unitext 92, Springer, Cham, 2016,https:\/\/doi.org\/10.1007\/978-3-319-15431-2.","DOI":"10.1007\/978-3-319-15431-2"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1061\/AJRUA6.0000816"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.21314\/JOR.2000.038"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/S0378-4266(02)00271-6"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1115\/1.4037623"},{"key":"atypb20","unstructured":"T. Takhtaganov,\n                      Efficient Estimation of Coherent Risk Measures for Risk-Averse Optimization Problems Governed by Partial Differential Equations with Random Inputs\n                      , Ph.D. thesis, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 2017."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2017.02.030"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/18M1220996"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1287\/ijoc.2015.0684"},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"Z. Zou, D. P. Kouri, and W. Aquino,\n                      An adaptive sampling approach for solving PDEs with uncertain inputs and evaluating risk\n                      , in Proceedings of the 19th AIAA Non-Deterministic Approaches Conference, AIAA SciTech Forum (AIAA 2017-1325), 2017,https:\/\/doi.org\/10.2514\/6.2017-1325.","DOI":"10.2514\/6.2017-1325"},{"key":"atypb25","doi-asserted-by":"crossref","unstructured":"Z. Zou, D. P. Kouri, and W. Aquino,\n                      A locally adapted reduced basis method for solving risk-averse PDE-constrained optimization problems\n                      , in Proceedings of the 2018 AIAA Non-Deterministic Approaches Conference (Kissimmee, FL), AIAA SciTech Forum (AIAA 2018-2174), 2018,https:\/\/doi.org\/10.2514\/6.2018-2174.","DOI":"10.2514\/6.2018-2174"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2018.10.028"}],"container-title":["SIAM\/ASA Journal on Uncertainty Quantification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1257433","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:22:18Z","timestamp":1787340138000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1257433"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":26,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/19M1257433"],"URL":"https:\/\/doi.org\/10.1137\/19m1257433","relation":{},"ISSN":["2166-2525"],"issn-type":[{"value":"2166-2525","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}