{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,12]],"date-time":"2024-09-12T19:11:44Z","timestamp":1726168304847},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2012,2,11]],"date-time":"2012-02-11T00:00:00Z","timestamp":1328918400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2012,10]]},"DOI":"10.1007\/s10589-012-9464-0","type":"journal-article","created":{"date-parts":[[2012,2,10]],"date-time":"2012-02-10T16:06:03Z","timestamp":1328889963000},"page":"591-617","source":"Crossref","is-referenced-by-count":14,"title":["Reduced order solution of structured linear systems arising in certain PDE-constrained optimization problems"],"prefix":"10.1007","volume":"53","author":[{"given":"V.","family":"Simoncini","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2012,2,11]]},"reference":[{"key":"9464_CR1","volume-title":"Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide","year":"2000","unstructured":"Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van\u00a0der Vorst, H. (eds.): Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia (2000)"},{"key":"9464_CR2","unstructured":"Bai, Z.-Z., Benzi, M., Chen, F., Wang, Z.-Q.: Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems. Technical Report 2011-001, Math\/CS Department, Emory University (January 2011). To appear in IMA J. Numer. Anal."},{"key":"9464_CR3","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/s10444-011-9173-8","volume":"35","author":"M. Benzi","year":"2011","unstructured":"Benzi, M., Haber, E., Taralli, L.: A preconditioning technique for a class of PDE-constrained optimization problems. Adv. Comput. Math. 35, 149\u2013173 (2011)","journal-title":"Adv. Comput. Math."},{"key":"9464_CR4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0962492904000212","volume":"14","author":"M. Benzi","year":"2005","unstructured":"Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1\u2013137 (2005)","journal-title":"Acta Numer."},{"issue":"2\u20133","key":"9464_CR5","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s10589-006-9001-0","volume":"36","author":"L. Bergamaschi","year":"2007","unstructured":"Bergamaschi, L., Gondzio, J., Venturin, M., Zilli, G.: Inexact constraint preconditioners for linear systems arising in interior point methods. Comput. Optim. Appl. 36(2\u20133), 137\u2013147 (2007)","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"9464_CR6","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1137\/060671590","volume":"51","author":"A. Borz\u00ec","year":"2009","unstructured":"Borz\u00ec, A., Schulz, V.: Multigrid methods for PDE optimization. SIAM Rev. 51(2), 361\u2013395 (2009)","journal-title":"SIAM Rev."},{"issue":"1","key":"9464_CR7","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1002\/nme.2758","volume":"82","author":"J. Boyle","year":"2010","unstructured":"Boyle, J., Mihajlovi\u0107, M.D., Scott, J.A.: HSL_MI20: an efficient AMG preconditioner. Int. J. Numer. Methods Eng., 82(1), 64\u201398 (2010)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"9464_CR8","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/0096-3003(86)90095-0","volume":"19","author":"A. Brandt","year":"1986","unstructured":"Brandt, A.: Algebraic multigrid theory: the symmetric case. Appl. Math. Comput. 19, 23\u201356 (1986)","journal-title":"Appl. Math. Comput."},{"key":"9464_CR9","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1002\/nla.572","volume":"15","author":"Z.-H. Cao","year":"2008","unstructured":"Cao, Z.-H.: Augmentation block preconditioners for saddle point-type matrices with singular (1, 1) blocks. Numer. Linear Algebra Appl. 15, 515\u2013533 (2008)","journal-title":"Numer. Linear Algebra Appl."},{"key":"9464_CR10","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1137\/05063427X","volume":"28","author":"H.S. Dollar","year":"2006","unstructured":"Dollar, H.S., Gould, N.I.M., Schilders, W.H.A., Wathen, A.J.: Implicit-factorization preconditioning and iterative solvers for regularized saddle-point systems. SIAM J. Matrix Anal. Appl. 28, 170\u2013189 (2006)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9464_CR11","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1002\/nla.308","volume":"10","author":"C. Durazzi","year":"2003","unstructured":"Durazzi, C., Ruggiero, V.: Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems. Numer. Linear Algebra Appl. 10, 673\u2013688 (2003)","journal-title":"Numer. Linear Algebra Appl."},{"key":"9464_CR12","series-title":"Numerical Mathematics and Scientific Computation","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198528678.001.0001","volume-title":"Finite Elements and Fast Iterative Solvers, with Applications in Incompressible Fluid Dynamics","author":"H.C. Elman","year":"2005","unstructured":"Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers, with Applications in Incompressible Fluid Dynamics. Numerical Mathematics and Scientific Computation, vol.\u00a021. Oxford University Press, London (2005)"},{"key":"9464_CR13","doi-asserted-by":"crossref","first-page":"1152","DOI":"10.1137\/080733413","volume":"31","author":"N. Gould","year":"2009","unstructured":"Gould, N., Simoncini, V.: Spectral analysis of saddle point matrices with indefinite leading blocks. SIAM J. Matrix Anal. Appl. 31, 1152\u20131171 (2009)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9464_CR14","doi-asserted-by":"crossref","first-page":"1376","DOI":"10.1137\/S1064827598345667","volume":"23","author":"N.I.M. Gould","year":"2001","unstructured":"Gould, N.I.M., Hribar, M.E., Nocedal, J.: On the solution of equality constrained quadratic programming problems arising in optimization. SIAM J. Sci. Comput. 23, 1376\u20131395 (2001)","journal-title":"SIAM J. Sci. Comput."},{"key":"9464_CR15","first-page":"114","volume":"22","author":"C. Greif","year":"2006","unstructured":"Greif, C., Sch\u00f6tzau, D.: Preconditioners for saddle point linear systems with highly singular (1, 1) blocks. Electron. Trans. Numer. Anal. 22, 114\u2013121 (2006)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"9464_CR16","first-page":"307","volume":"37","author":"C. Greif","year":"2010","unstructured":"Greif, C., Overton, M.: An analysis of low-rank modifications of preconditioners for saddle point systems. Electron. Trans. Numer. Anal. 37, 307\u2013320 (2010)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"9464_CR17","doi-asserted-by":"crossref","first-page":"1847","DOI":"10.1088\/0266-5611\/17\/6\/319","volume":"17","author":"E. Haber","year":"2001","unstructured":"Haber, E., Ascher, U.: Preconditioned all-at-once methods for large, sparse parameter estimation problems. Inverse Probl. 17, 1847\u20131864 (2001)","journal-title":"Inverse Probl."},{"issue":"5","key":"9464_CR18","doi-asserted-by":"crossref","first-page":"2291","DOI":"10.1137\/090779127","volume":"31","author":"R. Herzog","year":"2010","unstructured":"Herzog, R., Sachs, E.: Preconditioned conjugate gradient method for optimal control problems with control and state constraints. SIAM J. Matrix Anal. Appl. 31(5), 2291\u20132317 (2010)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"1","key":"9464_CR19","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1002\/nla.717","volume":"18","author":"P. Krzyzanowski","year":"2011","unstructured":"Krzyzanowski, P.: On block preconditioners for saddle point problems with singular or indefinite (1,\u00a01) block. Numer. Linear Algebra Appl. 18(1), 123\u2013140 (2011)","journal-title":"Numer. Linear Algebra Appl."},{"key":"9464_CR20","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1002\/(SICI)1099-1506(199805\/06)5:3<219::AID-NLA134>3.0.CO;2-7","volume":"5","author":"L. Luk\u0161an","year":"1998","unstructured":"Luk\u0161an, L., Vl\u010dek, J.: Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems. Numer. Linear Algebra Appl. 5, 219\u2013247 (1998)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"1","key":"9464_CR21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/nla.716","volume":"18","author":"K.-A. Mardal","year":"2011","unstructured":"Mardal, K.-A., Winther, R.: Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebra Appl. 18(1), 1\u201340 (2011)","journal-title":"Numer. Linear Algebra Appl."},{"key":"9464_CR22","unstructured":"The MathWorks, Inc. MATLAB 7, September 2004"},{"key":"9464_CR23","unstructured":"Oberle, H.J., Grimm, W.: BNDSCO-A Program for the numerical solution of optimal control problems. Technical report, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen (1989)"},{"key":"9464_CR24","doi-asserted-by":"crossref","first-page":"2754","DOI":"10.1137\/100792652","volume":"31","author":"M. Olshanskii","year":"2010","unstructured":"Olshanskii, M., Simoncini, V.: Acquired clustering properties and solution of certain saddle point systems. SIAM J. Matrix Anal. Appl. 31, 2754\u20132768 (2010)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9464_CR25","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1002\/1099-1506(200010\/12)7:7\/8<585::AID-NLA214>3.0.CO;2-F","volume":"7","author":"I. Perugia","year":"2000","unstructured":"Perugia, I., Simoncini, V.: Block\u2013diagonal and indefinite symmetric preconditioners for mixed finite element formulations. Numer. Linear Algebra Appl. 7, 585\u2013616 (2000)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"2","key":"9464_CR26","first-page":"341","volume":"46","author":"T. Rees","year":"2010","unstructured":"Rees, T., Stoll, M., Wathen, A.: All-at-once preconditioning in PDE-constrained optimization. Kybernetika 46(2), 341\u2013360 (2010)","journal-title":"Kybernetika"},{"key":"9464_CR27","doi-asserted-by":"crossref","first-page":"2903","DOI":"10.1137\/100798491","volume":"33","author":"T. Rees","year":"2011","unstructured":"Rees, T., Wathen, A.J.: Preconditioning iterative methods for the optimal control of the Stokes equations. SIAM J. Sci. Comput. 33, 2903\u20132926 (2011)","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"9464_CR28","doi-asserted-by":"crossref","first-page":"1992","DOI":"10.1137\/060661673","volume":"29","author":"T. Rees","year":"2007","unstructured":"Rees, T., Greif, C.: A preconditioner for linear systems arising from interior point optimization methods. SIAM J. Sci. Comput. 29(5), 1992\u20132007 (2007)","journal-title":"SIAM J. Sci. Comput."},{"key":"9464_CR29","doi-asserted-by":"crossref","unstructured":"Rees, T.: Preconditioning iterative methods for PDE constrained optimization. Ph.D. thesis, University of Oxford (2010)","DOI":"10.1002\/nla.693"},{"issue":"6","key":"9464_CR30","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1002\/nla.693","volume":"17","author":"T. Rees","year":"2010","unstructured":"Rees, T., Stoll, M.: Block-triangular preconditioners for PDE-constrained optimization. Numer. Linear Algebra Appl. 17(6), 977\u2013996 (2010)","journal-title":"Numer. Linear Algebra Appl."},{"key":"9464_CR31","volume-title":"Multigrid Methods, Frontiers Appl. Math.","author":"J.W. Ruge","year":"1987","unstructured":"Ruge, J.W., St\u00fcben, K.: Algebraic multigrid. In: Multigrid Methods, Frontiers Appl. Math. SIAM, Philadelphia (1987)"},{"key":"9464_CR32","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1137\/0914028","volume":"14","author":"Y. Saad","year":"1993","unstructured":"Saad, Y.: A flexible inner-outer preconditioned GMRES. SIAM J. Sci. Comput. 14, 461\u2013469 (1993)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"9464_CR33","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."},{"key":"9464_CR34","unstructured":"Sesana, D., Simoncini, V.: Spectral analysis of inexact constraint preconditioning for symmetric saddle point matrices. Technical report, Dipartimento di Matematica, Universit\u00e0 di Bologna (2010)"},{"key":"9464_CR35","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1023\/B:BITN.0000025094.68655.c7","volume":"44","author":"D. Silvester","year":"2004","unstructured":"Silvester, D., Mihajlovic, M.: A black-box multigrid preconditioner for the biharmonic equation. BIT Numer. Math. 44, 151\u2013163 (2004)","journal-title":"BIT Numer. Math."},{"key":"9464_CR36","doi-asserted-by":"crossref","first-page":"1352","DOI":"10.1137\/0731070","volume":"31","author":"D. Silvester","year":"1994","unstructured":"Silvester, D., Wathen, A.: Fast iterative solution of stabilized Stokes systems part II: using general block preconditioners. SIAM J. Numer. Anal. 31, 1352\u20131367 (1994)","journal-title":"SIAM J. Numer. Anal."},{"key":"9464_CR37","unstructured":"Simoncini, V.: Solution of structured algebraic linear systems in PDE-constrained optimization problems. Available at http:\/\/www.dm.unibo.it\/~simoncin\/ , July 2010. Slides of the talk given at the \u201cErice 2010 Workshop on Nonlinear Optimization, Variational Inequalities and Equilibrium Problems, 2\u201310 July 2010\u201d"},{"issue":"1","key":"9464_CR38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/nla.499","volume":"14","author":"V. Simoncini","year":"2007","unstructured":"Simoncini, V., Szyld, D.B.: Recent computational developments in Krylov subspace methods for linear systems. Numer. Linear Algebra Appl. 14(1), 1\u201359 (2007)","journal-title":"Numer. Linear Algebra Appl."},{"key":"9464_CR39","unstructured":"Stoll, M., Wathen, A.: All-at-once solution of time-dependent PDE-constrained optimization problems. Technical Report 1017, The Mathematical Institute, University of Oxford (2010)"},{"key":"9464_CR40","unstructured":"Thorne, H.S.: Distributed control and constraint preconditioners. Technical Report RAL-TR-2010-016, Rutherford Appleton Laboratory (2010)"},{"key":"9464_CR41","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/112","volume-title":"Optimal Control of Partial Differential Equations: Theory, Methods and Applications","author":"F. Tr\u00f6ltzsch","year":"2010","unstructured":"Tr\u00f6ltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Am. Math. Soc., Providence (2010). Translated by J. Sprekels"},{"key":"9464_CR42","first-page":"125","volume":"34","author":"A. Wathen","year":"2008\u20132009","unstructured":"Wathen, A., Rees, T.: Chebyshev semi-iteration in preconditioning for problems including the mass matrix. Electron. Trans. Numer. Anal. 34, 125\u2013135 (2008\u20132009)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"9464_CR43","doi-asserted-by":"crossref","first-page":"536","DOI":"10.1137\/100814767","volume":"32","author":"W. Zulehner","year":"2011","unstructured":"Zulehner, W.: Nonstandard norms and robust estimates for saddle point problems. SIAM. J. Matrix Anal. Appl. 32, 536 (2011)","journal-title":"SIAM. J. Matrix Anal. Appl."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-012-9464-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10589-012-9464-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-012-9464-0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,19]],"date-time":"2024-04-19T12:25:11Z","timestamp":1713529511000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10589-012-9464-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,2,11]]},"references-count":43,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2012,10]]}},"alternative-id":["9464"],"URL":"https:\/\/doi.org\/10.1007\/s10589-012-9464-0","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,2,11]]}}}