{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:47:44Z","timestamp":1747198064158,"version":"3.40.5"},"reference-count":34,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100002848","name":"Comisi\u00f3n Nacional de Investigaci\u00f3n Cient\u00edfica y Tecnol\u00f3gica","doi-asserted-by":"publisher","award":["1210579","1210391"],"award-info":[{"award-number":["1210579","1210391"]}],"id":[{"id":"10.13039\/501100002848","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations.\nWe complement the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm, and a Lagrange multiplier.\nOur final formulation fulfills the Babu\u0161ka\u2013Brezzi conditions on the continuous as well as discrete level, without restrictions.\nConsequently, we can allow for final-time desired states, and obtain an a posteriori error estimator which is efficient and reliable up to an additional discretization error of the adjoint problem.\nNumerical experiments confirm our theoretical findings.<\/jats:p>","DOI":"10.1515\/cmam-2023-0087","type":"journal-article","created":{"date-parts":[[2024,4,23]],"date-time":"2024-04-23T17:16:58Z","timestamp":1713892618000},"page":"673-691","source":"Crossref","is-referenced-by-count":2,"title":["Space-Time Least-Squares Finite Element Methods for Parabolic Distributed Optimal Control Problems"],"prefix":"10.1515","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5034-6593","authenticated-orcid":false,"given":"Thomas","family":"F\u00fchrer","sequence":"first","affiliation":[{"name":"Facultad de Matem\u00e1ticas , 28033 Pontificia Universidad Cat\u00f3lica de Chile , Santiago , Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-2001-4433","authenticated-orcid":false,"given":"Michael","family":"Karkulik","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica , 28090 Universidad T\u00e9cnica Federico Santa Mar\u00eda , Valpara\u00edso , Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2024,4,24]]},"reference":[{"key":"2024070116104986151_j_cmam-2023-0087_ref_001","doi-asserted-by":"crossref","unstructured":"R. Andreev,\nStability of sparse space-time finite element discretizations of linear parabolic evolution equations,\nIMA J. Numer. Anal. 33 (2013), no. 1, 242\u2013260.","DOI":"10.1093\/imanum\/drs014"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_002","doi-asserted-by":"crossref","unstructured":"R. Andreev,\nSpace-time discretization of the heat equation,\nNumer. Algorithms 67 (2014), no. 4, 713\u2013731.","DOI":"10.1007\/s11075-013-9818-4"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_003","doi-asserted-by":"crossref","unstructured":"J. Bey,\nSimplicial grid refinement: On Freudenthal\u2019s algorithm and the optimal number of congruence classes,\nNumer. Math. 85 (2000), no. 1, 1\u201329.","DOI":"10.1007\/s002110050475"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_004","doi-asserted-by":"crossref","unstructured":"P. B. Bochev and M. D. Gunzburger,\nLeast-Squares Finite Element Methods,\nAppl. Math. Sci. 166,\nSpringer, New York, 2009.","DOI":"10.1007\/b13382"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_005","doi-asserted-by":"crossref","unstructured":"D. Boffi, F. Brezzi and M. Fortin,\nMixed Finite Element Methods and Applications,\nSpringer Ser. Comput. Math. 44,\nSpringer, Heidelberg, 2013.","DOI":"10.1007\/978-3-642-36519-5"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_006","doi-asserted-by":"crossref","unstructured":"A. Borz\u00ec and V. Schulz,\nComputational Optimization of Systems Governed by Partial Differential Equations,\nComput. Sci. Eng. 8,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2012.","DOI":"10.1137\/1.9781611972054"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_007","unstructured":"R. Dautray and J.-L. Lions,\nMathematical Analysis and Numerical Methods for Science and Technology. Vol. 5,\nSpringer, Berlin, 1992."},{"key":"2024070116104986151_j_cmam-2023-0087_ref_008","doi-asserted-by":"crossref","unstructured":"D. Devaud and C. Schwab,\nSpace-time \n                  \n                     \n                        \n                           h\n                           \u2062\n                           p\n                        \n                     \n                     \n                     hp\n                  \n               -approximation of parabolic equations,\nCalcolo 55 (2018), no. 3, Paper No. 35.","DOI":"10.1007\/s10092-018-0275-2"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_009","doi-asserted-by":"crossref","unstructured":"L. Diening and J. Storn,\nA space-time DPG method for the heat equation,\nComput. Math. Appl. 105 (2022), 41\u201353.","DOI":"10.1016\/j.camwa.2021.11.013"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_010","doi-asserted-by":"crossref","unstructured":"L. Diening, J. Storn and T. Tscherpel,\nInterpolation operator on negative Sobolev spaces,\nMath. Comp. 92 (2023), no. 342, 1511\u20131541.","DOI":"10.1090\/mcom\/3824"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_011","unstructured":"L. C. Evans,\nPartial Differential Equations, 2nd ed.,\nGrad. Stud. Math. 19,\nAmerican Mathematical Society, Providence, 2010."},{"key":"2024070116104986151_j_cmam-2023-0087_ref_012","doi-asserted-by":"crossref","unstructured":"T. F\u00fchrer and M. Karkulik,\nSpace-time least-squares finite elements for parabolic equations,\nComput. Math. Appl. 92 (2021), 27\u201336.","DOI":"10.1016\/j.camwa.2021.03.004"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_013","doi-asserted-by":"crossref","unstructured":"T. F\u00fchrer and M. Karkulik,\nLeast-squares finite elements for distributed optimal control problems,\nNumer. Math. 154 (2023), no. 3\u20134, 409\u2013442.","DOI":"10.1007\/s00211-023-01367-7"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_014","doi-asserted-by":"crossref","unstructured":"G. Gantner and R. Stevenson,\nFurther results on a space-time FOSLS formulation of parabolic PDEs,\nESAIM Math. Model. Numer. Anal. 55 (2021), no. 1, 283\u2013299.","DOI":"10.1051\/m2an\/2020084"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_015","doi-asserted-by":"crossref","unstructured":"G. Gantner and R. Stevenson,\nA well-posed first order system least squares formulation of the instationary Stokes equations,\nSIAM J. Numer. Anal. 60 (2022), no. 3, 1607\u20131629.","DOI":"10.1137\/21M1432600"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_016","doi-asserted-by":"crossref","unstructured":"G. Gantner and R. Stevenson,\nApplications of a space-time FOSLS formulation for parabolic PDEs,\nIMA J. Numer. Anal. 44 (2024), no. 1, 58\u201382.","DOI":"10.1093\/imanum\/drad012"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_017","doi-asserted-by":"crossref","unstructured":"G. Gantner and R. Stevenson,\nImproved rates for a space-time FOSLS of parabolic PDEs,\nNumer. Math. 156 (2024), no. 1, 133\u2013157.","DOI":"10.1007\/s00211-023-01387-3"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_018","doi-asserted-by":"crossref","unstructured":"W. Gong, M. Hinze and Z. J. Zhou,\nSpace-time finite element approximation of parabolic optimal control problems,\nJ. Numer. Math. 20 (2012), no. 2, 111\u2013145.","DOI":"10.1515\/jnum-2012-0005"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_019","doi-asserted-by":"crossref","unstructured":"S. G\u00f6tschel and M. L. Minion,\nAn efficient parallel-in-time method for optimization with parabolic PDEs,\nSIAM J. Sci. Comput. 41 (2019), no. 6, C603\u2013C626.","DOI":"10.1137\/19M1239313"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_020","doi-asserted-by":"crossref","unstructured":"M. D. Gunzburger and A. Kunoth,\nSpace-time adaptive wavelet methods for optimal control problems constrained by parabolic evolution equations,\nSIAM J. Control Optim. 49 (2011), no. 3, 1150\u20131170.","DOI":"10.1137\/100806382"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_021","doi-asserted-by":"crossref","unstructured":"U. Langer, R. L\u00f6scher, O. Steinbach and H. Yang,\nAn adaptive finite element method for distributed elliptic optimal control problems with variable energy regularization,\nComput. Math. Appl. 160 (2024), 1\u201314.","DOI":"10.1016\/j.camwa.2024.02.006"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_022","doi-asserted-by":"crossref","unstructured":"U. Langer, O. Steinbach, F. Tr\u00f6ltzsch and H. Yang,\nSpace-time finite element discretization of parabolic optimal control problems with energy regularization,\nSIAM J. Numer. Anal. 59 (2021), no. 2, 675\u2013695.","DOI":"10.1137\/20M1332980"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_023","doi-asserted-by":"crossref","unstructured":"U. Langer, O. Steinbach, F. Tr\u00f6ltzsch and H. Yang,\nUnstructured space-time finite element methods for optimal control of parabolic equations,\nSIAM J. Sci. Comput. 43 (2021), no. 2, A744\u2013A771.","DOI":"10.1137\/20M1330452"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_024","unstructured":"U. Langer, O. Steinbach and H. Yang,\nRobust space-time finite element error estimates for parabolic distributed optimal control problems with energy regularization, preprint (2022), https:\/\/arxiv.org\/abs\/2206.06455."},{"key":"2024070116104986151_j_cmam-2023-0087_ref_025","doi-asserted-by":"crossref","unstructured":"J.-L. Lions,\nOptimal Control of Systems Governed by Partial Differential Equations,\nGrundlehren Math. Wiss. 170,\nSpringer, New York, 1971.","DOI":"10.1007\/978-3-642-65024-6"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_026","doi-asserted-by":"crossref","unstructured":"D. Meidner and B. Vexler,\nAdaptive space-time finite element methods for parabolic optimization problems,\nSIAM J. Control Optim. 46 (2007), no. 1, 116\u2013142.","DOI":"10.1137\/060648994"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_027","doi-asserted-by":"crossref","unstructured":"D. Meidner and B. Vexler,\nA priori error estimates for space-time finite element discretization of parabolic optimal control problems. I. Problems without control constraints,\nSIAM J. Control Optim. 47 (2008), no. 3, 1150\u20131177.","DOI":"10.1137\/070694016"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_028","doi-asserted-by":"crossref","unstructured":"M. Neum\u00fcller and E. Karabelas,\nGenerating admissible space-time meshes for moving domains in \n                  \n                     \n                        \n                           (\n                           \n                              d\n                              +\n                              1\n                           \n                           )\n                        \n                     \n                     \n                     (d+1)\n                  \n                dimensions,\nSpace-Time Methods\u2014Applications to Partial Differential Equations,\nRadon Ser. Comput. Appl. Math. 25,\nDe Gruyter, Berlin (2019), 185\u2013206.","DOI":"10.1515\/9783110548488-006"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_029","doi-asserted-by":"crossref","unstructured":"C. Schwab and R. Stevenson,\nSpace-time adaptive wavelet methods for parabolic evolution problems,\nMath. Comp. 78 (2009), no. 267, 1293\u20131318.","DOI":"10.1090\/S0025-5718-08-02205-9"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_030","doi-asserted-by":"crossref","unstructured":"E. M. Stein,\nSingular Integrals and Differentiability Properties of Functions,\nPrinceton Math. Ser. 30,\nPrinceton University, Princeton, 1970.","DOI":"10.1515\/9781400883882"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_031","doi-asserted-by":"crossref","unstructured":"O. Steinbach,\nSpace-time finite element methods for parabolic problems,\nComput. Methods Appl. Math. 15 (2015), no. 4, 551\u2013566.","DOI":"10.1515\/cmam-2015-0026"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_032","doi-asserted-by":"crossref","unstructured":"R. Stevenson and J. Westerdiep,\nStability of Galerkin discretizations of a mixed space-time variational formulation of parabolic evolution equations,\nIMA J. Numer. Anal. 41 (2021), no. 1, 28\u201347.","DOI":"10.1093\/imanum\/drz069"},{"key":"2024070116104986151_j_cmam-2023-0087_ref_033","unstructured":"V. Thom\u00e9e,\nGalerkin Finite Element Methods for Parabolic Problems, 2nd ed.,\nSpringer Ser. Comput. Math. 25,\nSpringer, Berlin, 2006."},{"key":"2024070116104986151_j_cmam-2023-0087_ref_034","doi-asserted-by":"crossref","unstructured":"F. Tr\u00f6ltzsch,\nOptimal Control of Partial Differential Equations,\nGrad. Stud. Math. 112,\nAmerican Mathematical Society, Providence, 2010.","DOI":"10.1090\/gsm\/112"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0087\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0087\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T16:14:22Z","timestamp":1719850462000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0087\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,24]]},"references-count":34,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,6,26]]},"published-print":{"date-parts":[[2024,7,1]]}},"alternative-id":["10.1515\/cmam-2023-0087"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2023-0087","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2024,4,24]]}}}