{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:47:22Z","timestamp":1787320042390,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>We consider evolution problems, such as diffusion convection equations or the linearized Navier\u2013Stokes system, or a weak coupling of them, which we would like to \u201cpredict\u201d on a time interval $(T_{0},T_{0}+T)$ but for which, on one hand, the initial value of the state variable is unknown. On the other hand \u201cmeasurements\u201d of the solutions are known on a time interval $(0,T_{0})$ and, for example, on a subdomain in the space variable. The classical approach in variational data assimilation is to look for the initial value at time 0, and this is known to be an ill-posed problem which has to be regularized. Here we propose to look for the value of the state variable at time $T_{0}$ (the end time of the \u201cmeasurements\u201d) and we prove on some basic examples that this is a well-posed problem. We give a result of exact reconstruction of the value at $T_{0}$ which is based on global Carleman inequalities, and we give an approximation algorithm which uses classical optimal control auxiliary problems. Using the same mathematical arguments, we also show why Tychonov regularization for variational data assimilation works in practical situations corresponding to realistic applications.<\/jats:p>","DOI":"10.1137\/060670961","type":"journal-article","created":{"date-parts":[[2009,3,11]],"date-time":"2009-03-11T21:31:17Z","timestamp":1236807077000},"page":"1089-1111","source":"Crossref","is-referenced-by-count":16,"title":["A Nonstandard Approach to a Data Assimilation Problem and Tychonov Regularization Revisited"],"prefix":"10.1137","volume":"48","author":[{"given":"Jean-Pierre","family":"Puel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,3,11]]},"reference":[{"key":"R1","first-page":"33","volume":"67","author":"Blum J.","year":"2002","journal-title":"Matapli"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1256\/smsqj.47812"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2002047"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"H. W. Engl, M. Hanke, and A. Neubauer,\n                      Regularization of Inverse Problems,\n                      Kluwer Academic Publishers, Dordrecht, 1996.","DOI":"10.1007\/978-94-009-1740-8"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/5\/4\/007"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1080\/03605309608821198"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2004.02.010"},{"key":"R8","unstructured":"A. Fursikov and O. Imanuvilov,\n                      Controllability of Evolution Equations,\n                      Lecture Notes Ser. 34, RIM-GARC, Seoul National University, Seoul, 1996."},{"key":"R9","unstructured":"G. C. Garcia, A. Osses, and J.P. Puel,\n                      A data assimilation problem applied to a large-scale ocean circulation model,\n                      submitted."},{"key":"R10","unstructured":"C. W. Groetsch,\n                      The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind,\n                      Res. Notes Math. 105, Pitman, Boston, 1984."},{"key":"R11","unstructured":"P. Hepperger,\n                      Ein POD-basierter Ansatz zur numerischen L\u00f6sung eines Datenassimilationsproblems am Beispiel einer linearen Diffusions-Konvektions-Gleichung,\n                      Diplomarbeit, T.U. M\u00fcnchen, M\u00fcnchen, Deutschland, 2007."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:1998104"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2001103"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.2977\/prims\/1145476103"},{"key":"R15","unstructured":"F. X. Le Dimet,\n                      Une \u00e9tude g\u00e9n\u00e9rale d'analyse objective variationnelle des champs m\u00e9t\u00e9orologiques,\n                      Rapport scientifique LAMP 28, Universit\u00e9 de Clermont II, Aubi\u00e8re, France, 1980."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1111\/j.1600-0870.1986.tb00459.x"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"J.L. Lions,\n                      Optimal Control of Systems Governed by Partial Differential Equations,\n                      Springer-Verlag, Berlin, 1971.","DOI":"10.1007\/978-3-642-65024-6"},{"key":"R18","unstructured":"J.L. Lions,\n                      Quelques m\u00e9thodes de r\u00e9solution des probl\u00e8mes aux limites non lin\u00e9aires,\n                      Dunod, Gauthier-Villars, Paris, 1969."},{"key":"R19","first-page":"219","volume":"31","author":"Mizohata S.","year":"1958","journal-title":"Mem. Coll. Sci. Univ. Kyoto Ser. A"},{"key":"R20","unstructured":"J.P. Puel,\n                      Application of Global Carleman Estimates to Controllability and Inverse Problems,\n                      Universidad Federal de Rio de Janeiro, Rio de Janeiro, Brazil, to appear."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/S1631-073X(02)02390-7"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"R. Temam,\n                      Navier\u2013Stokes Equations. Theory and Numerical Analysis.\n                      Stud. Math. Appl. 2, North\u2013Holland, Amsterdam, 1977.","DOI":"10.1115\/1.3424338"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060670961","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:54:51Z","timestamp":1787316891000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060670961"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/060670961"],"URL":"https:\/\/doi.org\/10.1137\/060670961","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}