{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,5]],"date-time":"2026-09-05T06:42:19Z","timestamp":1788590539840,"version":"build-2803163510"},"reference-count":42,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and\/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and\/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh\u2013Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.<\/jats:p>","DOI":"10.1137\/090766498","type":"journal-article","created":{"date-parts":[[2010,9,7]],"date-time":"2010-09-07T18:10:25Z","timestamp":1283883025000},"page":"2737-2764","source":"Crossref","is-referenced-by-count":1622,"title":["Nonlinear Model Reduction via Discrete Empirical Interpolation"],"prefix":"10.1137","volume":"32","author":[{"given":"Saifon","family":"Chaturantabut","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Danny C.","family":"Sorensen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,9,7]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"P. 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Willcox, and T. Backx,\n                      Missing point estimation in models described by proper orthogonal decomposition\n                      , in Proceedings of the 43rd IEEE Conference on Decision and Control (CDC 2004), Vol. 2, 2004, pp. 1767\u20131772.","DOI":"10.1109\/CDC.2004.1430301"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2008.2006102"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(02)00116-2"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2004.08.006"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1002\/pamm.200810057"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.2514\/1.2159"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2009.10.004"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1002\/nme.2453"},{"key":"R12","unstructured":"S. Chaturantabut and D. C. Sorensen,\n                      Application of POD and DEIM to dimension reduction of nonlinear miscible viscous fingering in porous media\n                      , Math. Comput. Model. Dyn. Syst., to appear."},{"key":"R13","doi-asserted-by":"crossref","unstructured":"S. Chaturantabut and D. C. Sorensen,\n                      Discrete empirical interpolation for nonlinear model reduction\n                      , in Proceedings of the 48th IEEE Conference on Decision and Control and the 28th Chinese Control Conference (CDC\/CCC 2009), 2009, pp. 4316\u20134321.","DOI":"10.1109\/CDC.2009.5400045"},{"key":"R14","unstructured":"Y. Chen,\n                      Model Order Reduction for Nonlinear Systems\n                      , Master's thesis, Massachusetts Institute of Technology, Cambridge, MA, 1999."},{"key":"R15","unstructured":"Y. Chen and J. White,\n                      A quadratic method for nonlinear model order reduction\n                      , in Technical Proceedings of the 2000 International Conference on Modeling and Simulation of Microsystems, 2000, pp. 477\u2013480."},{"key":"R16","doi-asserted-by":"crossref","unstructured":"N. Dong and J. Roychowdhury,\n                      Piecewise polynomial nonlinear model reduction\n                      , in Proceedings of the Design Automation Conference, IEEE Computer Society, Los Alamitos, CA, 2003, pp. 484\u2013489.","DOI":"10.1145\/775832.775957"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1002\/nme.2746"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"W. R. Graham, J. Peraire, and K. Y. Tang,\n                      Optimal control of vortex shedding using low-order models. Part\n                      I\u2014\n                      Open-loop model development\n                      , Internat. J. Numer. 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Yang,\n                      ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods\n                      , SIAM, Philadelphia, 1998.","DOI":"10.1137\/1.9780898719628"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(00)00270-6"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1023\/A:1015145924517"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1002\/nme.2086"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1002\/nme.2309"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1007\/s10092-009-0005-x"},{"key":"R30","unstructured":"A. T. Patera and G. J. Rozza,\n                      Reduced Basis Methods and A Posteriori Error Estimation for Parametrized Partial Differential Equations\n                      , MIT, Cambridge, MA, 2006-08, MIT Pappalardo Graduate Monographs in Mechanical Engineering, to appear; available online from http:\/\/mathicse.epfl.ch\/~rozza\/publications.html."},{"key":"R31","doi-asserted-by":"crossref","unstructured":"J. R. Phillips,\n                      Projection frameworks for model reduction of weakly nonlinear systems\n                      , in DAC '00: Proceedings of the 37th Annual Design Automation Conference, ACM, New York, 2000, pp. 184\u2013189.","DOI":"10.1145\/337292.337380"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1115\/1.1448332"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1002\/1097-0363(20001115)34:5<425::AID-FLD67>3.0.CO;2-W"},{"key":"R34","unstructured":"M. J. 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