{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T04:54:24Z","timestamp":1772340864403,"version":"3.50.1"},"reference-count":44,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["NFN S117-03"],"award-info":[{"award-number":["NFN S117-03"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,1,1]]},"abstract":"Abstract<\/jats:title>\n We consider elliptic distributed optimal control problems with energy\nregularization. Here the standard \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n {L_{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-norm regularization is\nreplaced by the \n \n \n \n H<\/m:mi>\n \n -<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n <\/m:math>\n \n {H^{-1}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-norm leading to more focused controls.\nIn this case, the optimality system can be reduced to a single singularly\nperturbed diffusion-reaction equation known as differential filter in\nturbulence theory. We investigate the error between\nthe finite element approximation \n \n \n \n u<\/m:mi>\n \n \u03f1<\/m:mi>\n \u2062<\/m:mo>\n h<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:math>\n \n {u_{\\varrho h}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> to the state u<\/jats:italic> and the\ndesired state \n \n \n \n u<\/m:mi>\n \u00af<\/m:mo>\n <\/m:mover>\n <\/m:math>\n \n {\\overline{u}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in terms of the mesh-size h<\/jats:italic> and the\nregularization parameter \u03f1. The choice \n \n \n \n \u03f1<\/m:mi>\n =<\/m:mo>\n \n h<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:mrow>\n <\/m:math>\n \n {\\varrho=h^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> ensures\noptimal convergence the rate of which only depends\non the regularity of the target function \n \n \n \n u<\/m:mi>\n \u00af<\/m:mo>\n <\/m:mover>\n <\/m:math>\n \n {\\overline{u}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nThe resulting symmetric and positive definite system of finite element\nequations is solved by the conjugate gradient (CG) method preconditioned\nby algebraic multigrid (AMG) or balancing domain decomposition by\nconstraints (BDDC). We numerically study robustness and efficiency of the\nAMG preconditioner with respect to h<\/jats:italic>, \u03f1, and the number of\nsubdomains (cores) p<\/jats:italic>. Furthermore, we investigate the parallel\nperformance of the BDDC preconditioned CG solver.<\/jats:p>","DOI":"10.1515\/cmam-2021-0169","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T20:50:27Z","timestamp":1633985427000},"page":"97-111","source":"Crossref","is-referenced-by-count":12,"title":["Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization"],"prefix":"10.1515","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3797-7475","authenticated-orcid":false,"given":"Ulrich","family":"Langer","sequence":"first","affiliation":[{"name":"Institute for Computational Mathematics , Johannes Kepler University Linz , Altenberger Stra\u00dfe 69, 4040 Linz , Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2552-3022","authenticated-orcid":false,"given":"Olaf","family":"Steinbach","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Angewandte Mathematik , Technische Universit\u00e4t Graz , Steyrergasse 30, 8010 Graz , Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huidong","family":"Yang","sequence":"additional","affiliation":[{"name":"Johann Radon Institute for Computational and Applied Mathematics , Austrian Academy of Sciences, Altenberger Stra\u00dfe 69, 4040 Linz , Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2021,10,10]]},"reference":[{"key":"2023033111454401030_j_cmam-2021-0169_ref_001","doi-asserted-by":"crossref","unstructured":"S. Badia, A. F. Mart\u00edn and J. Principe,\nImplementation and scalability analysis of balancing domain decomposition methods,\nArch. Comput. Methods Eng. 20 (2013), no. 3, 239\u2013262.","DOI":"10.1007\/s11831-013-9086-4"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_002","doi-asserted-by":"crossref","unstructured":"S. Badia, A. F. Mart\u00edn and J. Principe,\nMultilevel balancing domain decomposition at extreme scales,\nSIAM J. Sci. Comput. 38 (2016), no. 1, C22\u2013C52.","DOI":"10.1137\/15M1013511"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_003","unstructured":"L. C. Berselli, T. Iliescu and W. J. Layton,\nMathematics of Large Eddy Simulation of Turbulent Flows,\nSci. Comput.,\nSpringer, Berlin, 2006."},{"key":"2023033111454401030_j_cmam-2021-0169_ref_004","doi-asserted-by":"crossref","unstructured":"D. Braess,\nFinite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd ed.,\nCambridge University, Cambridge, 2007.","DOI":"10.1017\/CBO9780511618635"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_005","unstructured":"A. Brandt, S. McCormick and J. Ruge,\nAlgebraic multigrid (AMG) for sparse matrix equations,\nSparsity and its Applications,\nCambridge University, Cambridge (1985), 257\u2013284."},{"key":"2023033111454401030_j_cmam-2021-0169_ref_006","doi-asserted-by":"crossref","unstructured":"W. L. Briggs, V. E. Henson and S. F. McCormick,\nA Multigrid Tutorial, 2nd ed.,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719505"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_007","doi-asserted-by":"crossref","unstructured":"E. Casas,\nA review on sparse solutions in optimal control of partial differential equations,\nSeMA J. 74 (2017), no. 3, 319\u2013344.","DOI":"10.1007\/s40324-017-0121-5"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_008","doi-asserted-by":"crossref","unstructured":"C. R. Dohrmann,\nA preconditioner for substructuring based on constrained energy minimization,\nSIAM J. Sci. Comput. 25 (2003), no. 1, 246\u2013258.","DOI":"10.1137\/S1064827502412887"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_009","unstructured":"H. Goering, A. Felgenhauer, G. Lube, H.-G. Roos and L. Tobiska,\nSingularly Perturbed Differential Equations,\nMath. Res. 13,\nAkademie-Verlag, Berlin, 1983."},{"key":"2023033111454401030_j_cmam-2021-0169_ref_010","doi-asserted-by":"crossref","unstructured":"G. Haase and U. Langer,\nMultigrid methods: From geometrical to algebraic versions,\nModern Methods in Scientific Computing and Applications,\nSpringer, Dordrecht (2002), 103\u2013153.","DOI":"10.1007\/978-94-010-0510-4_4"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_011","doi-asserted-by":"crossref","unstructured":"W. Hackbusch,\nMulti-Grid Methods and Applications,\nSpringer, Heidelberg, 1985.","DOI":"10.1007\/978-3-662-02427-0"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_012","doi-asserted-by":"crossref","unstructured":"V. John,\nFinite Element Methods for Incompressible Flow Problems,\nSpringer Ser. Comput. Math. 51,\nSpringer, Cham, 2016.","DOI":"10.1007\/978-3-319-45750-5"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_013","doi-asserted-by":"crossref","unstructured":"G. Karypis and V. Kumar,\nA fast and high quality multilevel scheme for partitioning irregular graphs,\nSIAM J. Sci. Comput. 20 (1998), no. 1, 359\u2013392.","DOI":"10.1137\/S1064827595287997"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_014","doi-asserted-by":"crossref","unstructured":"S. Kaya and C. C. Manica,\nConvergence analysis of the finite element method for a fundamental model in turbulence,\nMath. Models Methods Appl. Sci. 22 (2012), no. 11, Article ID 1250033.","DOI":"10.1142\/S0218202512500339"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_015","doi-asserted-by":"crossref","unstructured":"F. Kickinger,\nAlgebraic multi-grid for discrete elliptic second-order problems,\nMultigrid Methods V,\nLect. Notes Comput. Sci. Eng. 3,\nSpringer, Berlin (1998), 157\u2013172.","DOI":"10.1007\/978-3-642-58734-4_9"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_016","doi-asserted-by":"crossref","unstructured":"J. Kraus and M. Wolfmayr,\nOn the robustness and optimality of algebraic multilevel methods for reaction-diffusion type problems,\nComput. Vis. Sci. 16 (2013), no. 1, 15\u201332.","DOI":"10.1007\/s00791-014-0221-z"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_017","doi-asserted-by":"crossref","unstructured":"U. Langer, M. Neum\u00fcller and A. Schafelner,\nSpace-time finite element methods for parabolic evolution problems with variable coefficients,\nAdvanced Finite Element Methods with Applications,\nSpringer, Cham (2019), 247\u2013275.","DOI":"10.1007\/978-3-030-14244-5_13"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_018","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":"2023033111454401030_j_cmam-2021-0169_ref_019","doi-asserted-by":"crossref","unstructured":"U. Langer and H. Yang,\nRobust and efficient monolithic fluid-structure-interaction solvers,\nInternat. J. 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Madden,\nBoundary layer preconditioners for finite-element discretizations of singularly perturbed reaction-diffusion problems,\nNumer. Algorithms 79 (2018), no. 1, 281\u2013310.","DOI":"10.1007\/s11075-017-0437-3"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_030","doi-asserted-by":"crossref","unstructured":"T. A. Nhan and N. Madden,\nAn analysis of diagonal and incomplete Cholesky preconditioners for singularly perturbed problems on layer-adapted meshes,\nJ. Appl. Math. Comput. 65 (2021), no. 1\u20132, 245\u2013272.","DOI":"10.1007\/s12190-020-01390-z"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_031","doi-asserted-by":"crossref","unstructured":"M. A. Olshanskii and A. Reusken,\nOn the convergence of a multigrid method for linear reaction-diffusion problems,\nComputing 65 (2000), no. 3, 193\u2013202.","DOI":"10.1007\/s006070070006"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_032","unstructured":"C. Popa,\nAlgebraic multigrid smoothing property of Kaczmarz\u2019s relaxation for general rectangular linear systems,\nElectron. Trans. Numer. Anal. 29 (2007\/08), 150\u2013162."},{"key":"2023033111454401030_j_cmam-2021-0169_ref_033","doi-asserted-by":"crossref","unstructured":"J. W. Ruge and K. St\u00fcben,\nAlgebraic multigrid,\nMultigrid Methods,\nFrontiers Appl. Math. 3,\nSIAM, Philadelphia (1987), 73\u2013130.","DOI":"10.1137\/1.9781611971057.ch4"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_034","doi-asserted-by":"crossref","unstructured":"A. H. Schatz and L. B. Wahlbin,\nOn the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions,\nMath. Comp. 40 (1983), no. 161, 47\u201389.","DOI":"10.1090\/S0025-5718-1983-0679434-4"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_035","doi-asserted-by":"crossref","unstructured":"G. Stadler,\nElliptic optimal control problems with \n \n \n \n L\n 1\n \n \n \n L^{1}\n \n -control cost and applications for the placement of control devices,\nComput. Optim. Appl. 44 (2009), no. 2, 159\u2013181.","DOI":"10.1007\/s10589-007-9150-9"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_036","doi-asserted-by":"crossref","unstructured":"O. Steinbach,\nNumerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-68805-3"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_037","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":"2023033111454401030_j_cmam-2021-0169_ref_038","doi-asserted-by":"crossref","unstructured":"O. Steinbach and H. Yang,\nComparison of algebraic multigrid methods for an adaptive space-time finite-element discretization of the heat equation in 3D and 4D,\nNumer. Linear Algebra Appl. 25 (2018), no. 3, Article ID e2143.","DOI":"10.1002\/nla.2143"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_039","doi-asserted-by":"crossref","unstructured":"A. Toselli and O. Widlund,\nDomain Decomposition Methods\u2014Algorithms and Theory,\nSpringer Ser. Comput. Math. 34,\nSpringer, Berlin, 2005.","DOI":"10.1007\/b137868"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_040","doi-asserted-by":"crossref","unstructured":"F. Tr\u00f6ltzsch,\nOptimal Control of Partial Differential Equations: Theory, Methods and Applications,\nGrad. Stud. Math. 112,\nAmerican Mathematical Society, Providence, 2010.","DOI":"10.1090\/gsm\/112\/07"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_041","doi-asserted-by":"crossref","unstructured":"R. Verf\u00fcrth,\nA Posteriori Error Esimtation Techniques for Finite Element Methods,\nOxford University, Oxford, 2013.","DOI":"10.1093\/acprof:oso\/9780199679423.001.0001"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_042","doi-asserted-by":"crossref","unstructured":"J. Xu and L. Zikatanov,\nAlgebraic multigrid methods,\nActa Numer. 26 (2017), 591\u2013721.","DOI":"10.1017\/S0962492917000083"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_043","doi-asserted-by":"crossref","unstructured":"H. Yang and W. Zulehner,\nNumerical simulation of fluid-structure interaction problems on hybrid meshes with algebraic multigrid methods,\nJ. Comput. Appl. Math. 235 (2011), no. 18, 5367\u20135379.","DOI":"10.1016\/j.cam.2011.05.046"},{"key":"2023033111454401030_j_cmam-2021-0169_ref_044","doi-asserted-by":"crossref","unstructured":"S. Zampini and X. Tu,\nMultilevel balancing domain decomposition by constraints deluxe algorithms with adaptive coarse spaces for flow in porous media,\nSIAM J. Sci. 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