{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:42:37Z","timestamp":1787330557074,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>In this paper, we describe block matrix algorithms for the iterative solution of a large-scale linear-quadratic optimal control problem involving a parabolic partial differential equation over a finite control horizon. We consider an \u201call at once\u201d discretization of the problem and formulate three iterative algorithms. The first algorithm is based on preconditioning a symmetric positive definite reduced linear system involving only the unknown control variables; however inner-outer iterations are required. The second algorithm modifies the first algorithm to avoid inner-outer iterations by introducing an auxiliary variable. It yields a symmetric indefinite system with a positive definite block preconditioner. The third algorithm is the central focus of this paper. It modifies the preconditioner in the second algorithm by a parallel-in-time preconditioner based on the parareal algorithm. Theoretical results show that the preconditioned algorithms have optimal convergence properties and parallel scalability. Numerical experiments confirm the theoretical results.<\/jats:p>","DOI":"10.1137\/080717481","type":"journal-article","created":{"date-parts":[[2010,4,16]],"date-time":"2010-04-16T18:09:44Z","timestamp":1271441384000},"page":"1180-1200","source":"Crossref","is-referenced-by-count":48,"title":["Analysis of Block Parareal Preconditioners for Parabolic Optimal Control Problems"],"prefix":"10.1137","volume":"32","author":[{"given":"Tarek P.","family":"Mathew","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marcus","family":"Sarkis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christian E.","family":"Schaerer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,4,16]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"A. Battermann and M. Heinkenschloss,\n                      Preconditioners for Karush-Kuhn-Tucker matrices arising in the optimal control of distributed systems\n                      , in Optimal Control of Partial Differential Equations, W. Desch, F. Kappel, and K. Kunisch, eds., Birkh\u00e4user, Basel, 1998, pp. 15\u201332.","DOI":"10.1007\/978-3-0348-8849-3_2"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"A. Battermann and E. W. Sachs,\n                      Block preconditioner for KKT systems in PDE-governed optimal control problems\n                      , in Workshop on Fast Solutions of Discretized Optimization Problems, H. W. Hoppe, K.H. Hoffmann, and V. Schulz, eds., Birkh\u00e4user, Basel, 2001, pp. 1\u201318.","DOI":"10.1007\/978-3-0348-8233-0_1"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000212"},{"key":"R4","unstructured":"M. Berggren and M. Heinkenschloss,\n                      Parallel solution of optimal-control problems by time-domain decomposition\n                      , in Computational Science for the 21st Century, M.O. Bristeau, G. Etgen, W. Fitzgibbon, J. L. Lions, J. Periaux, and M. F. Wheeler, eds., Wiley, Chichester, 1997, pp. 102\u2013112."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S106482750241565X"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827502415661"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"J. W. Demmel,\n                      Applied Numerical Linear Algebra\n                      , SIAM, Philadelphia, 1997.","DOI":"10.1137\/1.9781611971446"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1002\/nme.860"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"M. J. Gander and S. Vandewalle,\n                      On the super linear and linear convergence of the parareal algorithm\n                      , in Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng. 55, D. E. Keyes and O. B. Widlund, eds., Springer, New York, 2005, pp. 291\u2013298.","DOI":"10.1007\/978-3-540-34469-8_34"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/05064607X"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/17\/6\/319"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2004.03.005"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(00)01793-6"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"J. L. Lions,\n                      Optimal Control of Systems Governed by Partial Differential Equations\n                      , Springer, New York, 1971.","DOI":"10.1007\/978-3-642-65024-6"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"A. Locatelli,\n                      Optimal Control: An Introduction\n                      , Birkh\u00e4user, Berlin, 2001.","DOI":"10.1007\/978-3-0348-8328-3"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/S1631-073X(02)02467-6"},{"key":"R17","unstructured":"T. P. Mathew, M. Sarkis, and C. E. Schaerer,\n                      Block iterative algorithms for the solution of parabolic optimal control problems\n                      , in 7th International Meeting High Performance Computing for Computational Science - VECPAR 2006, Lecture Notes in Comput. Sci. 4395, Springer, New York, 2007, pp. 452\u2013465."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595288206"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1002\/nla.526"},{"key":"R20","unstructured":"P. Neittaan\u00e4ki and D. 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Varga,\n                      Matrix Iterative Analysis\n                      , Prentice Hall, Englewood Cliffs, NJ, 1962."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080717481","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:12:03Z","timestamp":1787328723000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080717481"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080717481"],"URL":"https:\/\/doi.org\/10.1137\/080717481","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}