{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,3]],"date-time":"2026-09-03T16:55:06Z","timestamp":1788454506864,"version":"build-2803163510"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, particularly in problems of design. The solution of such PDE-constrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. The large-dimensional linear systems which result from discretization and which need to be solved are of saddle-point type. We introduce two optimal preconditioners for these systems, which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other PDEs.<\/jats:p>","DOI":"10.1137\/080727154","type":"journal-article","created":{"date-parts":[[2010,2,5]],"date-time":"2010-02-05T18:13:52Z","timestamp":1265393632000},"page":"271-298","source":"Crossref","is-referenced-by-count":180,"title":["Optimal Solvers for PDE-Constrained Optimization"],"prefix":"10.1137","volume":"32","author":[{"given":"Tyrone","family":"Rees","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H. Sue","family":"Dollar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrew J.","family":"Wathen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,2,5]]},"reference":[{"key":"R1","first-page":"1","volume":"15","author":"Asher U. M.","year":"2003","journal-title":"Electron. Trans. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1097-4067","issn-type":"print"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1002\/nla.469"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000212"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"G. Biros and G. Dogan,\n                      A multilevel algorithm for inverse problems with elliptic PDE constraints\n                      , Inverse Problems, 24 (2008), paper 034010.","DOI":"10.1088\/0266-5611\/24\/3\/034010"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S106482750241565X"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/060671590"},{"key":"R7","unstructured":"J. Boyle, M. D. Mihajlovic, and J. A. Scott,\n                      HSL_MI$20$: An Efficient AMG Preconditioner\n                      , Technical Report RAL-TR-2007-021, Department of Computational and Applied Mathematics, Rutherford Appleton Laboratory, Chilton, UK, 2007."},{"key":"R8","unstructured":"A. Brandt, S. McCormick, and J. Ruge,\n                      Algebraic multigrid (AMG) for sparse matrix equations\n                      , in Sparsity and Its Applications, D. J. Evans, ed., Cambridge University Press, Cambridge, UK, 1985, pp. 257\u2013284."},{"key":"R9","doi-asserted-by":"crossref","unstructured":"W. L. Briggs, V. E. Henson, and S. F. McCormick,\n                      A Multigrid Tutorial\n                      , 2nd ed., SIAM, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719505"},{"key":"R10","unstructured":"S. S. Collis and M. Heinkenschloss,\n                      Analysis of the Streamline Upwind\/Petrov Galerkin Method Applied to the Solution of Optimal Control Problems\n                      , Technical Report TR02\u201301, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 2002."},{"key":"R11","doi-asserted-by":"crossref","unstructured":"H. Elman, D. Silvester, and A. Wathen,\n                      Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics\n                      , Numer. Math. Sci. Comput., Oxford University Press, Oxford, UK, 2005.","DOI":"10.1093\/oso\/9780198528678.001.0001"},{"key":"R12","unstructured":"M. Engel and M. Griebel,\n                      A Multigrid Method for Constrained Optimal Control Problems\n                      , Technical report, SFB-Preprint 406, Sonderforschungsbereich 611, Rheinische Friedrich-Wilhelms-Universit\u00e4t Bonn, Bonn, Germany, 2008."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01386013"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827598345667"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/17\/6\/319"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"W. Hackbush,\n                      Multi-Grid Methods and Applications\n                      , Springer-Verlag, New York, 1985.","DOI":"10.1007\/978-3-662-02427-0"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/040612774"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/070681910"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012994261707"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899351805"},{"key":"R21","unstructured":"J. L. Lions,\n                      Optimal Control of Systems\n                      , Springer, New York, 1968."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008725519350"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008774521095"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827599355153"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/0712047"},{"key":"R26","unstructured":"T. Rees, H. Dollar, and A. Wathen,\n                      Optimal Solvers for PDE Constrained Optimization\n                      , Technical Report TR 08\/10, Computing Laboratory, University of Oxford, Oxford, UK, 2008."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1137\/060660977"},{"key":"R28","unstructured":"U. Trottenberg, C. Oosterlee, and A. Sch\u00fcller,\n                      Multigrid\n                      , Academic Press, New York, 2000."},{"key":"R29","unstructured":"R. Varga,\n                      Matrix Iterative Analysis\n                      , Prentice\u2013Hall, Englewood Cliffs, NJ, 1962."},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/7.4.449"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080727154","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:50:33Z","timestamp":1787334633000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080727154"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":30,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080727154"],"URL":"https:\/\/doi.org\/10.1137\/080727154","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}