{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T09:27:22Z","timestamp":1780565242532,"version":"3.54.1"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2011,3,1]],"date-time":"2011-03-01T00:00:00Z","timestamp":1298937600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput. Visual Sci."],"published-print":{"date-parts":[[2011,3]]},"DOI":"10.1007\/s00791-011-0168-2","type":"journal-article","created":{"date-parts":[[2011,11,17]],"date-time":"2011-11-17T14:21:07Z","timestamp":1321539667000},"page":"131-141","source":"Crossref","is-referenced-by-count":30,"title":["Convergence analysis of multigrid methods with collective point smoothers for optimal control problems"],"prefix":"10.1007","volume":"14","author":[{"given":"Stefan","family":"Takacs","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Walter","family":"Zulehner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2011,11,18]]},"reference":[{"key":"168_CR1","unstructured":"Arian, E., Ta\u2019asan, S.: Multigrid One-Shot Methods for Optimal Control Problems: Infinite Dimensional Control (1994)"},{"key":"168_CR2","unstructured":"Battermann, A., Heinkenschloss, M.: Preconditioners for Karush\u2013Kuhn\u2013Tucker matrices arising in the optimal control of distributed systems. In: Desch, W., et\u00a0al. (eds.) Control and estimation of distributed"},{"key":"168_CR3","doi-asserted-by":"crossref","unstructured":"Battermann, A., Sachs, E.W.: Block preconditioners for KKT systems in PDE-governed optimal control problems. In: Hoffmann, K.-H., et\u00a0al. (eds.) Fast Solution of Discretized Optimization Problems. Workshop at the Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany, pp. 1\u201318. Birkh\u00e4user, Basel, 8\u201312 May 2000. ISNM, Int. Ser. Numer. Math., vol. 138 (2001)","DOI":"10.1007\/978-3-0348-8233-0_1"},{"issue":"2","key":"168_CR4","doi-asserted-by":"crossref","first-page":"714","DOI":"10.1137\/S1064827502415661","volume":"27","author":"G. Biros","year":"2005","unstructured":"Biros G., Ghattas O.: Parallel Lagrange\u2013Newton\u2013Krylov-Schur methods for PDE-constrained optimization. II: The Lagrange-Newton solver and its application to optimal control of steady viscous flows. SIAM J. Sci. Comput. 27(2), 714\u2013739 (2005)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"168_CR5","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1137\/S106482750241565X","volume":"27","author":"G. Biros","year":"2005","unstructured":"Biros G., Ghattas O.: Parallel Lagrange\u2013Newton\u2013Krylov-Schur methods for PDE-constrained optimization. I: The Krylov-Schur solver. SIAM J. Sci. Comput. 27(2), 687\u2013713 (2005)","journal-title":"SIAM J. Sci. Comput."},{"key":"168_CR6","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1137\/060671590","volume":"51","author":"A. Borzi","year":"2009","unstructured":"Borzi A., Schulz V.: Multigrid methods for PDE optimization. SIAM Rev. 51, 361\u2013395 (2009)","journal-title":"SIAM Rev."},{"issue":"5","key":"168_CR7","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1137\/S0363012901393432","volume":"41","author":"A. Borzi","year":"2003","unstructured":"Borzi A., Kunisch K., Kwak D.Y.: Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system. SIAM J. Control Optim. 41(5), 1477\u20131497 (2003)","journal-title":"SIAM J. Control Optim."},{"key":"168_CR8","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0086682","volume-title":"Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions. In: Lecture Notes in Mathematics vol 1341","author":"M. Dauge","year":"1988","unstructured":"Dauge M.: Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions. In: Lecture Notes in Mathematics vol 1341. Springer, Berlin (1988)"},{"key":"168_CR9","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1007\/BF01204238","volume":"15","author":"M. Dauge","year":"1992","unstructured":"Dauge M.: Neumann and mixed problems on curvilinear polyhedra. Integr. Equ. Oper. Theory 15, 227\u2013261 (1992)","journal-title":"Integr. Equ. Oper. Theory"},{"issue":"2","key":"168_CR10","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1002\/(SICI)1099-1506(199603\/04)3:2<161::AID-NLA76>3.0.CO;2-P","volume":"3","author":"A. Ecker","year":"1996","unstructured":"Ecker A., Zulehner W.: On the smoothing property of multi-grid methods in non-symmetric case. Numer. Lin. Alg. Appl. 3(2), 161\u2013172 (1996)","journal-title":"Numer. Lin. Alg. Appl."},{"key":"168_CR11","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1007\/BF00934478","volume":"31","author":"W. Hackbusch","year":"1980","unstructured":"Hackbusch W.: Fast solution of elliptic control problems. J. Optim. Theory Appl. 31, 565\u2013581 (1980)","journal-title":"J. Optim. Theory Appl."},{"key":"168_CR12","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-02427-0","volume-title":"Multi-Grid Methods and Applications","author":"W. Hackbusch","year":"1985","unstructured":"Hackbusch W.: Multi-Grid Methods and Applications. Springer, Berlin (1985)"},{"issue":"3","key":"168_CR13","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1023\/B:BITN.0000046815.96929.b8","volume":"44","author":"S.B. Hazra","year":"2004","unstructured":"Hazra S.B., Schulz V.: Simultaneous pseudo-timestepping for PDE-model based optimization problems. BIT 44(3), 457\u2013472 (2004)","journal-title":"BIT"},{"issue":"1\u20132","key":"168_CR14","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s00607-008-0024-5","volume":"84","author":"O. Lass","year":"2009","unstructured":"Lass O., Vallejos M., Borzi A., Douglas C.C.: Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems. Computing 84(1\u20132), 27\u201348 (2009)","journal-title":"Computing"},{"key":"168_CR15","first-page":"351","volume-title":"Les m\u00e9thodes directes en th\u00e9orie des \u00e9quations elliptiques","author":"J. Necas","year":"1967","unstructured":"Necas J.: Les m\u00e9thodes directes en th\u00e9orie des \u00e9quations elliptiques, pp. 351. Masson, Paris (1967)"},{"issue":"3","key":"168_CR16","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1307\/mmj\/1031732779","volume":"13","author":"C. Pearcy","year":"1966","unstructured":"Pearcy C.: An elementary proof of the power inequality for the numerical radius. Michigan Math. J. 13(3), 289\u2013291 (1966)","journal-title":"Michigan Math. J."},{"issue":"1","key":"168_CR17","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1137\/080727154","volume":"32","author":"T. Rees","year":"2010","unstructured":"Rees T., Dollar S., Wathen A.: Optimal solvers for pde-constrained optimization. SIAM J. Sci. Comput. 32(1), 271\u2013298 (2010)","journal-title":"SIAM J. Sci. Comput."},{"key":"168_CR18","unstructured":"Reusken, A.: A new lemma in multigrid convergence theory. Report-RANA 91-07 (1991)"},{"issue":"3","key":"168_CR19","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1137\/060660977","volume":"29","author":"J. Sch\u00f6berl","year":"2007","unstructured":"Sch\u00f6berl J., Zulehner W.: Symmetric indefinite preconditioners for saddle point problems with applications to pde-constrained optimization problems. SIAM J. Matrix Anal. Appl. 29(3), 752\u2013773 (2007)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"4","key":"168_CR20","doi-asserted-by":"crossref","first-page":"1482","DOI":"10.1137\/100783285","volume":"49","author":"J. Sch\u00f6berl","year":"2011","unstructured":"Sch\u00f6berl J., Simon R., Zulehner W.: A robust multigrid method for elliptic optimal control problems. SIAM J. Numer. Anal. 49(4), 1482\u20131503 (2011)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4-11","key":"168_CR21","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s00791-008-0094-0","volume":"11","author":"V. Schulz","year":"2008","unstructured":"Schulz V., Wittum G.: Transforming smoothers for PDE constrained optimization problems. Comput. Vis. Sci. 11(4-11), 207\u2013219 (2008)","journal-title":"Comput. Vis. Sci."},{"key":"168_CR22","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1007\/s00211-008-0187-1","volume":"111","author":"R. Simon","year":"2009","unstructured":"Simon R., Zulehner W.: On Schwarz-type smoothers for saddle point problems with applications to PDE-constrained optimization problems. Numer. Math. 111, 445\u2013468 (2009)","journal-title":"Numer. Math."},{"key":"168_CR23","unstructured":"Ta\u2019asan, S.: \u201dOne-shot\u201d methods for optimal control of distributed parameter systems I: the finite dimensional case. ICASE-Report 91-2, NASA Langley Research Center, Hampton, VA (1991)"},{"key":"168_CR24","volume-title":"Multigrid","author":"U. Trottenberg","year":"2001","unstructured":"Trottenberg U., Oosterlee C., Sch\u00fcller A.: Multigrid. Academic Press, London (2001)"},{"issue":"5","key":"168_CR25","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1007\/BF01396361","volume":"54","author":"G. Wittum","year":"1988","unstructured":"Wittum G.: Multi-grid methods for Stokes and Navier-Stokes equations. Transforming smoothers: algorithms and numerical results. Numer. Math. 54(5), 543\u2013563 (1988)","journal-title":"Numer. Math."},{"issue":"1","key":"168_CR26","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/BF01386394","volume":"57","author":"G. Wittum","year":"1990","unstructured":"Wittum G.: On the convergence of multi-grid methods with transforming smoothers. Numer. Math. 57(1), 15\u201338 (1990)","journal-title":"Numer. Math."}],"container-title":["Computing and Visualization in Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00791-011-0168-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00791-011-0168-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00791-011-0168-2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,19]],"date-time":"2019-06-19T15:04:11Z","timestamp":1560956651000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00791-011-0168-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,3]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2011,3]]}},"alternative-id":["168"],"URL":"https:\/\/doi.org\/10.1007\/s00791-011-0168-2","relation":{},"ISSN":["1432-9360","1433-0369"],"issn-type":[{"value":"1432-9360","type":"print"},{"value":"1433-0369","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,3]]}}}