{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T08:35:11Z","timestamp":1772786111385,"version":"3.50.1"},"reference-count":30,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,10,1]]},"abstract":"Abstract<\/jats:title>\n We consider standard tracking-type, distributed elliptic optimal control problems with \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n L^{2}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> regularization, and their finite element discretization.\nWe are investigating the \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n L^{2}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> error between the 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> of the state \n \n \n \n u<\/m:mi>\n \u03f1<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n u_{\\varrho}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and the desired state (target) \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 regularization parameter \ud835\udf1a and the mesh size \u210e that leads to the optimal choice \n \n \n \n \u03f1<\/m:mi>\n =<\/m:mo>\n \n h<\/m:mi>\n 4<\/m:mn>\n <\/m:msup>\n <\/m:mrow>\n <\/m:math>\n \n \\varrho=h^{4}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nIt turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble\u2013Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand.\nThe theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.<\/jats:p>","DOI":"10.1515\/cmam-2022-0138","type":"journal-article","created":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T13:09:28Z","timestamp":1676639368000},"page":"989-1005","source":"Crossref","is-referenced-by-count":7,"title":["Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems"],"prefix":"10.1515","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3797-7475","authenticated-orcid":false,"given":"Ulrich","family":"Langer","sequence":"first","affiliation":[{"name":"Institute of Computational Mathematics , Johannes Kepler University Linz , Altenberger Stra\u00dfe 69, 4040 Linz , Austria"}]},{"given":"Richard","family":"L\u00f6scher","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Angewandte Mathematik , Technische Universit\u00e4t Graz , Steyrergasse 30, 8010 Graz , Austria"}]},{"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"}]},{"given":"Huidong","family":"Yang","sequence":"additional","affiliation":[{"name":"Computational Science Center , Universit\u00e4t Wien , Oskar-Morgenstern-Platz 1, 1090 Wien , Austria"}]}],"member":"374","published-online":{"date-parts":[[2023,2,18]]},"reference":[{"key":"2025011504140378776_j_cmam-2022-0138_ref_001","doi-asserted-by":"crossref","unstructured":"O. Axelsson, M. Neytcheva and A. Str\u00f6m,\nAn efficient preconditioning method for state box-constrained optimal control problems,\nJ. Numer. Math. 26 (2018), no. 4, 185\u2013207.","DOI":"10.1515\/jnma-2017-0047"},{"key":"2025011504140378776_j_cmam-2022-0138_ref_002","doi-asserted-by":"crossref","unstructured":"M. Benzi, G. H. Golub and J. Liesen,\nNumerical solution of saddle point problems,\nActa Numer. 14 (2005), 1\u2013137.","DOI":"10.1017\/S0962492904000212"},{"key":"2025011504140378776_j_cmam-2022-0138_ref_003","doi-asserted-by":"crossref","unstructured":"A. Borzi and V. Schulz,\nMultigrid methods for PDE optimization,\nSIAM Rev. 51 (2009), no. 2, 361\u2013395.","DOI":"10.1137\/060671590"},{"key":"2025011504140378776_j_cmam-2022-0138_ref_004","doi-asserted-by":"crossref","unstructured":"J. H. Bramble and J. E. Pasciak,\nA preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems,\nMath. Comp. 50 (1988), no. 181, 1\u201317.","DOI":"10.1090\/S0025-5718-1988-0917816-8"},{"key":"2025011504140378776_j_cmam-2022-0138_ref_005","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\nThe Mathematical Theory of Finite Element Methods,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2025011504140378776_j_cmam-2022-0138_ref_006","doi-asserted-by":"crossref","unstructured":"H. C. Elman, D. J. Silvester and A. J. Wathen,\nFinite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics,\nNumer. Math. Sci. Comput.,\nOxford University, New York, 2005.","DOI":"10.1093\/oso\/9780198528678.001.0001"},{"key":"2025011504140378776_j_cmam-2022-0138_ref_007","doi-asserted-by":"crossref","unstructured":"A. Ern and J.-L. Guermond,\nTheory and Practice of Finite Elements,\nAppl. Math. 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