PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2007 | 27 | 1 | 95-117
Tytuł artykułu

Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Proper orthogonal decomposition (POD) is a powerful technique for model reduction of linear and non-linear systems. It is based on a Galerkin type discretization with basis elements created from the system itself. In this work, error estimates for Galerkin POD methods for linear elliptic, parameter-dependent systems are proved. The resulting error bounds depend on the number of POD basis functions and on the parameter grid that is used to generate the snapshots and to compute the POD basis. The error estimates also hold for semi-linear elliptic problems with monotone nonlinearity. Numerical examples are included.
Twórcy
  • Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
  • Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
Bibliografia
  • [1] H.W. Alt, Lineare Funktionalanalysis. Eine anwendungsorientierte Einführung, Springer-Verlag, Berlin 1992.
  • [2] J.A. Atwell and B.B. King, Reduced order controllers for spatially distributed systems via proper orthogonal decomposition, SIAM Journal Scientific Computation, to appear.
  • [3] H.T. Banks, M.L. Joyner, B. Winchesky and W.P. Winfree, Nondestructive evaluation using a reduced-order computational methodology, Inverse Problems 16 (2000), 1-17.
  • [4] M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations, C.R. Acad. Sci. Paris, Ser. I 339 (2004), 667-672.
  • [5] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, Rhode Island 1998.
  • [6] K. Fukuda, Introduction to Statistical Recognition, Academic Press, New York 1990.
  • [7] M.D. Gunzburger, L. Hou and T.P. Svobodny, Finite element approximations of an optimal control problem associated with the scalar Ginzburg-Landau equation, Computers Math. Applic. 21 (1991), 123-131.
  • [8] M. Hinze and S. Volkwein, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition, to appear in Computational Optimization and Applications.
  • [9] P. Holmes, J.L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge Monographs on Mechanics, Cambridge University Press 1996.
  • [10] C. Homescu, L.R. Petzold and R. Serban, Error estimation for reduced order models of dynamical systems, SIAM J. Numer. Anal. 43 (2005), 1693-1714.
  • [11] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin 1980.
  • [12] K. Kunisch and S. Volkwein, Control of Burgers' equation by a reduced order approach using proper orthogonal decomposition, J. Optimization Theory and Applications 102 (1999), 345-371.
  • [13] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numerische Mathematik 90 (2001), 117-148.
  • [14] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM Journal on Numerical Analysis 40 (2002), 492-515.
  • [15] S. Lall, J.E. Marsden and S. Glavaski, Empirical model reduction of controlled nonlinear systems, in: Proceedings of the IFAC Congress, vol. F (1999), 473-478.
  • [16] H.V. Ly and H.T. Tran, Modelling and control of physical processes using proper orthogonal decomposition, Mathematical and Computer Modeling 33 (2001), 223-236.
  • [17] L. Machiels, Y. Maday and A.T. Patera, Output bounds for reduced-order approximations of elliptic partial differential equations, Comput. Methods Appl. Mech. Engrg. 190 (2001), 3413-3426.
  • [18] Y. Maday and E.M. Rønquist, A reduced-basis element method, Journal of Scientific Computing 17 (2002), 1-4.
  • [19] M. Rathinam and L. Petzold, Dynamic iteration using reduced order models: a method for simulation of large scale modular systems, SIAM J. Numer. Anal. 40 (2002), 1446-1474.
  • [20] C.W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition, Int. J. on Bifurcation and Chaos 15 (2005), 997-1013.
  • [21] L. Sirovich, Turbulence and the dynamics of coherent structures, parts I-III, Quarterly of Applied Mathematics XLV (1987), 561-590.
  • [22] S. Volkwein, Boundary control of the Burgers equation: optimality conditions and reduced-order approach, in: K.-H. Hoffmann, I. Lasiecka, G. Leugering, J. Sprekels and F. Tröltzsch, eds, Optimal Control of Complex Structures, International Series of Numerical Mathematics 139 (2001), 267-278.
  • [23] K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition, AIAA (2001), 2001-2611.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1078
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.