Document Zbl 1346.65035

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Numer. Math.</a> <a href="/?q=in%3A358208" title="Articles in this Issue">108, 87-101 (2016)</a>. </div> <div class="abstract">Summary: In this manuscript we consider the development of fast iterative solvers for Stokes control problems, an important class of partial differential equation-constrained optimization problems. In particular we wish to develop effective preconditioners for the matrix systems arising from finite element discretizations of time-dependent variants of such problems. To do this we consider a suitable rearrangement of the matrix systems, and exploit the saddle point structure of many of the relevant sub-matrices involved – we may then use this to construct representations of these sub-matrices based on good approximations of their $(1, 1)$-block and Schur complement. We test our recommended iterative methods on a distributed control problem with Dirichlet boundary conditions, and on a time-periodic problem.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6617338">Cited in <strong>3</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65K10" title="MSC2020">65K10</a> </td> <td class="space"> Numerical optimization and variational techniques </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49J20" title="MSC2020">49J20</a> </td> <td class="space"> Existence theories for optimal control problems involving partial differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65F05" title="MSC2020">65F05</a> </td> <td class="space"> Direct numerical methods for linear systems and matrix inversion </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3APDE-constrained+optimization">PDE-constrained optimization</a>; <a href="/?q=ut%3Atime-dependent+Stokes+control">time-dependent Stokes control</a>; <a href="/?q=ut%3Apreconditioning">preconditioning</a>; <a href="/?q=ut%3Asaddle+point+system">saddle point system</a>; <a href="/?q=ut%3ASchur+complement">Schur complement</a>; <a href="/?q=ut%3Acommutator+argument">commutator argument</a>; <a href="/?q=ut%3Afast+iterative+solver">fast iterative solver</a>; <a href="/?q=ut%3Afinite+element">finite element</a>; <a href="/?q=ut%3ADirichlet+boundary+conditions">Dirichlet boundary conditions</a>; <a href="/?q=ut%3Atime-periodic+problem">time-periodic problem</a></div> <div class="software"> <h3>Software:</h3><a href="/software/8090">AGMG</a>; <a href="/software/7246">HSL_MI20</a>; <a href="/software/4398">IFISS</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1346.65035" data-ciurl="/ci/06617338" data-biburl="/bibtex/06617338.bib" data-amsurl="/amsrefs/06617338.bib" data-xmlurl="/xml/06617338.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06617338.pdf" title="Zbl 1346.65035 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.apnum.2016.05.002" aria-label="DOI for “Fast iterative solvers for large matrix systems arising from time-dependent Stokes control problems”" title="10.1016/j.apnum.2016.05.002">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://kar.kent.ac.uk/48160/7/TimeDeptStokes_Elsevier_Final.pdf" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Axelsson, O.; Neytcheva, M., Eigenvalue estimates for preconditioned saddle point matrices, Numer. Linear Alg. Appl., 13, 339-360 (2006) · <a href="/1224.65080" class="nowrap">Zbl 1224.65080</a></td> </tr><tr> <td>[2]</td> <td class="space">Boyle, J.; Mihajlovic, M. D.; Scott, J. A., HSL_MI20: an efficient AMG preconditioner for finite element problems in 3D, Int. J. Numer. Meth. Eng., 82, 64-98 (2010) · <a href="/1183.76799" class="nowrap">Zbl 1183.76799</a></td> </tr><tr> <td>[3]</td> <td class="space">Cahouet, J.; Chabard, J.-P., Some fast 3D finite element solvers for the generalized Stokes problem, Int. J. Numer. Meth. Fl., 8, 869-895 (1988) · <a href="/0665.76038" class="nowrap">Zbl 0665.76038</a></td> </tr><tr> <td>[4]</td> <td class="space">Elman, H. C.; Ramage, A.; Silvester, D. J., Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow, ACM T. Math. Software, 33, 2-14 (2007) · <a href="/1365.65326" class="nowrap">Zbl 1365.65326</a></td> </tr><tr> <td>[5]</td> <td class="space">Elman, H. C.; Ramage, A.; Silvester, D. J., IFISS: a computational laboratory for investigating incompressible flow problems, SIAM Review, 56, 261-273 (2014) · <a href="/1426.76645" class="nowrap">Zbl 1426.76645</a></td> </tr><tr> <td>[6]</td> <td class="space">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 Series (2005), Oxford University Press: Oxford University Press New York · <a href="/1083.76001" class="nowrap">Zbl 1083.76001</a></td> </tr><tr> <td>[8]</td> <td class="space">Ipsen, I. C.F., A note on preconditioning non-symmetric matrices, SIAM J. Sci. Comput., 23, 1050-1051 (2001) · <a href="/0998.65049" class="nowrap">Zbl 0998.65049</a></td> </tr><tr> <td>[9]</td> <td class="space">Kay, D.; Loghin, D.; Wathen, A., A preconditioner for the steady-state Navier-Stokes equations, SIAM J. Sci. Comput., 24, 237-256 (2002) · <a href="/1013.65039" class="nowrap">Zbl 1013.65039</a></td> </tr><tr> <td>[10]</td> <td class="space">Krendl, W.; Simoncini, V.; Zulehner, W., Efficient preconditioning for an optimal control problem with the time-periodic Stokes equations, (Lect. Notes Comput. Sci. Eng., vol. 103 (2015)), 479-487 · <a href="/1320.65100" class="nowrap">Zbl 1320.65100</a></td> </tr><tr> <td>[11]</td> <td class="space">Krendl, W.; Simoncini, V.; Zulehner, W., Stability estimates and structural spectral properties of saddle point problems, Numer. Math., 124, 183-213 (2013) · <a href="/1269.65032" class="nowrap">Zbl 1269.65032</a></td> </tr><tr> <td>[12]</td> <td class="space">Kuznetsov, Y. A., Efficient iterative solvers for elliptic finite element problems on nonmatching grids, Russ. J. Numer. Anal. M., 10, 187-211 (1995) · <a href="/0839.65031" class="nowrap">Zbl 0839.65031</a></td> </tr><tr> <td>[13]</td> <td class="space">Murphy, M. F.; Golub, G. H.; Wathen, A. J., A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21, 1969-1972 (2000) · <a href="/0959.65063" class="nowrap">Zbl 0959.65063</a></td> </tr><tr> <td>[14]</td> <td class="space">Napov, A.; Notay, Y., An algebraic multigrid method with guaranteed convergence rate, SIAM J. Sci. Comput., 34, A1079-A1109 (2012) · <a href="/1248.65037" class="nowrap">Zbl 1248.65037</a></td> </tr><tr> <td>[15]</td> <td class="space">Notay, Y., An aggregation-based algebraic multigrid method, Electron. Trans. Numer. Anal., 37, 123-146 (2010) · <a href="/1206.65133" class="nowrap">Zbl 1206.65133</a></td> </tr><tr> <td>[16]</td> <td class="space">Notay, Y., Aggregation-based algebraic multigrid for convection-diffusion equations, SIAM J. Sci. Comput., 34, A2288-A2316 (2012) · <a href="/1250.76139" class="nowrap">Zbl 1250.76139</a></td> </tr><tr> <td>[17]</td> <td class="space">Notay, Y., AGMG software and documentation, see</td> </tr><tr> <td>[18]</td> <td class="space">Paige, C. C.; Saunders, M. A., Solutions of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12, 617-629 (1975) · <a href="/0319.65025" class="nowrap">Zbl 0319.65025</a></td> </tr><tr> <td>[19]</td> <td class="space">Pearson, J., Fast iterative solvers for PDE-constrained optimization problems (2013), University of Oxford, D.Phil. thesis</td> </tr><tr> <td>[20]</td> <td class="space">Pearson, J. W., On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems, Electron. Trans. Numer. Anal., 44, 53-72 (2015) · <a href="/1312.65104" class="nowrap">Zbl 1312.65104</a></td> </tr><tr> <td>[21]</td> <td class="space">Pearson, J. W., Preconditioned iterative methods for Navier-Stokes control problems, J. Comp. Phys., 292, 194-207 (2015) · <a href="/1349.76087" class="nowrap">Zbl 1349.76087</a></td> </tr><tr> <td>[22]</td> <td class="space">Pearson, J. W.; Stoll, M., Fast iterative solution of reaction-diffusion control problems arising from chemical processes, SIAM J. Sci. Comput., 35, B987-B1009 (2013) · <a href="/1281.65095" class="nowrap">Zbl 1281.65095</a></td> </tr><tr> <td>[23]</td> <td class="space">Pearson, J. W.; Stoll, M.; Wathen, A. J., Regularization-robust preconditioners for time-dependent PDE-constrained optimization problems, SIAM J. Matrix Anal. Appl., 33, 1126-1152 (2012) · <a href="/1263.65035" class="nowrap">Zbl 1263.65035</a></td> </tr><tr> <td>[24]</td> <td class="space">Pearson, J. W.; Wathen, A. J., A new approximation of the Schur complement in preconditioners for PDE-constrained optimization, Numer. Linear Alg. Appl., 19, 816-829 (2012) · <a href="/1274.65187" class="nowrap">Zbl 1274.65187</a></td> </tr><tr> <td>[25]</td> <td class="space">Pearson, J. W.; Wathen, A. J., Fast iterative solvers for convection-diffusion control problems, Electron. Trans. Numer. Anal., 40, 294-310 (2013) · <a href="/1287.49031" class="nowrap">Zbl 1287.49031</a></td> </tr><tr> <td>[26]</td> <td class="space">Silvester, D.; Elman, H.; Ramage, A., Incompressible Flow and Iterative Solver Software (IFISS), version 3.3 (2014)</td> </tr><tr> <td>[27]</td> <td class="space">Silvester, D.; Wathen, A., Fast iterative solution of stabilised Stokes systems. Part II: using general block preconditioners, SIAM J. Numer. Anal., 31, 1352-1367 (1994) · <a href="/0810.76044" class="nowrap">Zbl 0810.76044</a></td> </tr><tr> <td>[28]</td> <td class="space">Stoll, M.; Wathen, A., All-at-once solution of time-dependent Stokes control, J. Comp. Phys., 232, 498-515 (2013)</td> </tr><tr> <td>[29]</td> <td class="space">Takacs, S., A robust all-at-once multigrid method for the Stokes control problem, Numer. Math., 130, 517-540 (2015) · <a href="/1329.76276" class="nowrap">Zbl 1329.76276</a></td> </tr><tr> <td>[30]</td> <td class="space">Tröltzsch, F., Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Series in Mathematics (2010), American Mathematical Society · <a href="/1195.49001" class="nowrap">Zbl 1195.49001</a></td> </tr><tr> <td>[31]</td> <td class="space">Wathen, A. J.; Rees, T., Chebyshev semi-iteration in preconditioning for problems including the mass matrix, Electron. Trans. Numer. Anal., 34, 125-135 (2009) · <a href="/1189.65065" class="nowrap">Zbl 1189.65065</a></td> </tr><tr> <td>[32]</td> <td class="space">Zulehner, W., Nonstandard norms and robust estimates for saddle point problems, SIAM J. Matrix Anal. Appl., 32, 536-560 (2011) · <a href="/1251.65078" class="nowrap">Zbl 1251.65078</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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