{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:35:12Z","timestamp":1787337312731,"version":"build-2736575974"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We discuss the computation of balanced truncation model reduction for a class of descriptor systems which include the semidiscrete Oseen equations with time-independent advection and the linearized Navier\u2013Stokes equations, linearized around a steady state. The purpose of this paper is twofold. First, we show how to apply standard balanced truncation model reduction techniques, which apply to dynamical systems given by ordinary differential equations, to this class of descriptor systems. This is accomplished by eliminating the algebraic equation using a projection. The second objective of this paper is to demonstrate how the important class of ADI\/Smith-type methods for the approximate computation of reduced order models using balanced truncation can be applied without explicitly computing the aforementioned projection. Instead, we utilize the solution of saddle point problems. We demonstrate the effectiveness of the technique in the computation of reduced order models for semidiscrete Oseen equations.<\/jats:p>","DOI":"10.1137\/070681910","type":"journal-article","created":{"date-parts":[[2008,3,5]],"date-time":"2008-03-05T19:21:22Z","timestamp":1204744882000},"page":"1038-1063","source":"Crossref","is-referenced-by-count":110,"title":["Balanced Truncation Model Reduction for a Class of Descriptor Systems with Application to the Oseen Equations"],"prefix":"10.1137","volume":"30","author":[{"given":"Matthias","family":"Heinkenschloss","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Danny C.","family":"Sorensen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kai","family":"Sun","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,3,5]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"K. Afanasiev and M. Hinze,\n                      Adaptive control of a wake flow using proper orthogonal decomposition\n                      , in Shape Optimization and Optimal Design (Cambridge, 1999), Lecture Notes in Pure and Appl. Math. 216, Dekker, New York, 2001, pp. 317\u2013332.","DOI":"10.1201\/9780203904169.ch13"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"A. C. Antoulas,\n                      Approximation of Large-Scale Systems\n                      , SIAM, Philadelphia, 2005.","DOI":"10.1137\/1.9780898718713"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"A. C. Antoulas, D. C. Sorensen, and S. Gugercin,\n                      A survey of model reduction methods for large-scale systems\n                      , in Structured Matrices in Mathematics, Computer Science, and Engineering, I (Boulder, CO, 1999), Contemp. Math. 280, American Mathematical Society, Providence, RI, 2001, pp. 193\u2013219.","DOI":"10.1090\/conm\/280\/04630"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"U. M. Ascher and L. R. Petzold,\n                      Computer Methods for Ordinary Differential Equations and Differential\u2013Algebraic Equations\n                      , SIAM, Philadelphia, 1998.","DOI":"10.1137\/1.9781611971392"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"P. Benner, V. Mehrmann, and D. C. Sorensen, eds.\n                      Dimension Reduction of Large-Scale Systems\n                      , Lect. Notes Comput. Sci. Eng. 45, Springer-Verlag, Heidelberg, 2005.","DOI":"10.1007\/3-540-27909-1"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"F. Brezzi and M. Fortin,\n                      Mixed and Hybrid Finite Element Methods\n                      , Computational Mathematics 15, Springer-Verlag, Berlin, 1991.","DOI":"10.1007\/978-1-4612-3172-1"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2006.10.026"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"R. F. Curtain,\n                      Model reduction for control design for distributed parameter systems\n                      , in Research Directions in Distributed Parameter Systems (Raleigh, NC, 2000), Frontiers Appl. Math. 27, SIAM, Philadelphia, 2003, pp. 95\u2013121.","DOI":"10.1137\/1.9780898717525.ch4"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"G. E. Dullerud and F. Paganini,\n                      A Course in Robust Control Theory\n                      , Texts Appl. Math. 36, Springer-Verlag, Berlin, Heidelberg, New York, 2000.","DOI":"10.1007\/978-1-4757-3290-0"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"H. C. Elman, D. J. Silvester, and A. J. Wathen,\n                      Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics\n                      , Oxford University Press, Oxford, 2005.","DOI":"10.1093\/oso\/9780198528678.001.0001"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"M. Fahl and E. Sachs,\n                      Reduced order modelling approaches to PDE\u2013constrained optimization based on proper orthogonal decompostion\n                      , in Large-Scale PDE-Constrained Optimization, Lect. Notes Comput. Sci. Eng. 30, L. T. Biegler, O. Ghattas, M. Heinkenschloss, and B. van Bloemen Waanders, eds., Springer-Verlag, Heidelberg, 2003.","DOI":"10.1007\/978-3-642-55508-4_16"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1080\/00207178408933239"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1080\/00207170410001713448"},{"key":"R14","unstructured":"S. Gugercin and J.R. Li,\n                      Smith-type methods for balanced truncation of large sparse systems\n                      , in Dimension Reduction of Large-Scale Systems, Lect. Notes Comput. Sci. Eng. 45, P. Benner, V. Mehrmann, and D. C. Sorensen, eds., Springer-Verlag, Heidelberg, 2005, pp. 83\u2013115."},{"key":"R15","doi-asserted-by":"crossref","unstructured":"M. D. Gunzburger,\n                      Finite Element Methods for Incompressible Viscous Flows: A Guide to Theory, Practice and Algorithms\n                      , Academic Press, Boston, 1989.","DOI":"10.1016\/B978-0-12-307350-1.50009-0"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"E. Hairer and G. Wanner,\n                      Solving Ordinary Differential Equations\n                      II.\n                      Stiff and Differential Algebraic Problems\n                      , Springer Ser. Comput. Math. 14, Springer-Verlag, Berlin, Heidelberg, New York, 1991.","DOI":"10.1007\/978-3-662-09947-6"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"P. Holmes, J. L. Lumley, and G. Berkooz,\n                      Turbulence, Coherent Structures, Dynamical Systems and Symmetry\n                      , Cambridge Monogr. Mech., Cambridge University Press, Cambridge, New York, 1996.","DOI":"10.1017\/CBO9780511622700"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.5943"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/050632774"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021732508059"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142900382612"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/j.paerosci.2003.12.001"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"V. Mehrmann and T. Stykel,\n                      Balanced truncation model reduction for large-scale systems in descriptor form\n                      , in Dimension Reduction of Large-Scale Systems, Lect. Notes Comput. Sci. Eng. 45, P. Benner, V. Mehrmann, and D. C. Sorensen, eds., Springer-Verlag, Heidelberg, 2005, pp. 83\u2013115.","DOI":"10.1007\/3-540-27909-1_3"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1981.1102568"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827598347666"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2006.01.007"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1137\/0910047"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1063\/1.1333038"},{"key":"R29","doi-asserted-by":"crossref","unstructured":"A. Quarteroni and A. Valli,\n                      Numerical Approximation of Partial Differential Equations\n                      , Springer-Verlag, Berlin, Heidelberg, New York, 1994.","DOI":"10.1007\/978-3-540-85268-1"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127405012429"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2003.03.001"},{"key":"R32","unstructured":"J. Sabino,\n                      Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method\n                      , Ph.D. thesis, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 2006. Available as CAAM TR06\u201308."},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2004.01.015"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1002\/fld.867"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1137\/0111073"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.2514\/2.1570"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070681910","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:56:21Z","timestamp":1787334981000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070681910"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":36,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/070681910"],"URL":"https:\/\/doi.org\/10.1137\/070681910","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}