{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T10:23:55Z","timestamp":1790331835804,"version":"4.1.0"},"reference-count":16,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,13]],"date-time":"2020-08-13T00:00:00Z","timestamp":1597276800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Minist\u00e8re de l'Enseignement sup\u00e9rieur, de la Recherche et de l'Innovation","award":["CDSN"],"award-info":[{"award-number":["CDSN"]}]},{"DOI":"10.13039\/501100013388","name":"Office National d'\u00e9tudes et de Recherches A\u00e9rospatiales","doi-asserted-by":"publisher","award":["LMA2S"],"award-info":[{"award-number":["LMA2S"]}],"id":[{"id":"10.13039\/501100013388","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001655","name":"Deutscher Akademischer Austauschdienst","doi-asserted-by":"publisher","award":["57442045"],"award-info":[{"award-number":["57442045"]}],"id":[{"id":"10.13039\/501100001655","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["415980428"],"award-info":[{"award-number":["415980428"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["390685689"],"award-info":[{"award-number":["390685689"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation probabilities over large classes of probability distributions, we present an adaptive algorithm for the reconstruction of increasing real-valued functions. While this problem is similar to the classical statistical problem of isotonic regression, the optimisation setting alters several characteristics of the problem and opens natural algorithmic possibilities. We present our algorithm, establish sufficient conditions for convergence of the reconstruction to the ground truth, and apply the method to synthetic test cases and a real-world example of uncertainty quantification for aerodynamic design.<\/jats:p>","DOI":"10.3390\/a13080196","type":"journal-article","created":{"date-parts":[[2020,8,13]],"date-time":"2020-08-13T09:23:44Z","timestamp":1597310624000},"page":"196","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Adaptive Reconstruction of Imperfectly Observed Monotone Functions, with Applications to Uncertainty Quantification"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5721-6089","authenticated-orcid":false,"given":"Luc","family":"Bonnet","sequence":"first","affiliation":[{"name":"ONERA, 29 Avenue de la Division Leclerc, 92320 Ch\u00e2tillon, France"},{"name":"Laboratoire MSSMat\u2014UMR CNRS 8579, CentraleSup\u00e9lec, 8\u201310 rue Joliot Curie, 91190 Gif sur Yvette, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Luc","family":"Akian","sequence":"additional","affiliation":[{"name":"ONERA, 29 Avenue de la Division Leclerc, 92320 Ch\u00e2tillon, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3767-0281","authenticated-orcid":false,"given":"\u00c9ric","family":"Savin","sequence":"additional","affiliation":[{"name":"ONERA, 29 Avenue de la Division Leclerc, 92320 Ch\u00e2tillon, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4081-8301","authenticated-orcid":false,"given":"T.","family":"Sullivan","sequence":"additional","affiliation":[{"name":"Mathematics Institute and School of Engineering, University of Warwick, Coventry CV4 7AL, UK"},{"name":"Zuse Institute Berlin, Takustra\u00dfe 7, 14195 Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,13]]},"reference":[{"key":"ref_1","unstructured":"Barlow, R.E., Bartholomew, D.J., Bremner, J.M., and Brunk, H.D. (1972). Statistical Interference under Order Restrictions. The Theory and Application of Isotonic Regression, John Wiley & Sons."},{"key":"ref_2","first-page":"1","article-title":"Isotone optimization in R: Pool-Adjacent-Violators Algorithm (PAVA) and active set methods","volume":"32","author":"Hornik","year":"2009","journal-title":"J. Stat. Softw."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1198\/TECH.2010.10111","article-title":"Nearly-isotonic regression","volume":"53","author":"Tibshirani","year":"2011","journal-title":"Technometrics"},{"key":"ref_4","unstructured":"Jordan, A.I., M\u00fchlemann, A., and Ziegel, J.F. (2019). Optimal solutions to the isotonic regression problem. arXiv."},{"key":"ref_5","unstructured":"Robertson, T., Wright, F.T., and Dykstra, R.L. (1988). Order Restricted Statistical Inference, John Wiley & Sons."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Groeneboom, P., and Jongbloed, G. (2014). Nonparametric Estimation under Shape Constraints: Estimators. Algorithms and Asymptotics, Cambridge University Press.","DOI":"10.1017\/CBO9781139020893"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1137\/10080782X","article-title":"Optimal Uncertainty Quantification","volume":"55","author":"Owhadi","year":"2013","journal-title":"SIAM Rev."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1657","DOI":"10.1051\/m2an\/2013083","article-title":"Optimal Uncertainty Quantification for legacy data observations of Lipschitz functions","volume":"47","author":"Sullivan","year":"2013","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1368","DOI":"10.1137\/13094712X","article-title":"Convex Optimal Uncertainty Quantification","volume":"25","author":"Han","year":"2015","journal-title":"SIAM J. Optim."},{"key":"ref_10","unstructured":"Cook, P.H., McDonald, M.A., and Firmin, M.C.P. (1979). Aerofoil RAE 2822 Pressure Distributions, and boundary layer and wake measurements. Experimental Data Base for Computer Program Assessment, NATO. AGARD Advisory Report No. 138."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1051\/meca\/2013056","article-title":"The ONERA elsA CFD software: Input from research and feedback from industry","volume":"14","author":"Cambier","year":"2013","journal-title":"Mech. Ind."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hirsch, C., Wunsch, D., Szumbarski, J., \u0141aniewski-Wo\u0142\u0142k, \u0141., and Pons-Prats, J. (2019). Ordinary kriging surrogates in aerodynamics. Uncertainty Management for Robust Industrial Design in Aeronautics: Findings and Best Practice Collected During UMRIDA, a Collaborative Research Project (2013\u20132016) Funded by the European Union, Springer International Publishing.","DOI":"10.1007\/978-3-319-77767-2"},{"key":"ref_13","unstructured":"Garner, H.C., Rogers, E.W.E., Acum, W.E.A., and Maskell, E.C. (1966). Subsonic Wind Tunnel Jwall Corrections, NATO. AGARDo-Graph 109."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Haase, W., Bradsma, F., Elsholz, E., Leschziner, M., and Schwamborn, D. (1993). EUROVAL\u2014An European Initiative on Validation of CFD Codes, Vieweg Verlag.","DOI":"10.1007\/978-3-663-14131-0"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1023\/A:1008202821328","article-title":"Differential Evolution\u2014A simple and efficient heuristic for global optimization over continuous spaces","volume":"11","author":"Storn","year":"1997","journal-title":"J. Glob. Optim."},{"key":"ref_16","unstructured":"van der Walt, S., and Millman, J. (2011, January 11\u201316). Building a framework for predictive science. Proceedings of the 10th Python in Science Conference (SciPy 2011), Austin, TX, USA."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/8\/196\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:00:16Z","timestamp":1760176816000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/8\/196"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,13]]},"references-count":16,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["a13080196"],"URL":"https:\/\/doi.org\/10.3390\/a13080196","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,13]]}}}