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Advances in Computational Mathematics

, Volume 41, Issue 5, pp 961–986 | Cite as

A nonintrusive reduced basis method applied to aeroacoustic simulations

  • Fabien CasenaveEmail author
  • Alexandre Ern
  • Tony Lelièvre
Article

Abstract

The Reduced Basis Method can be exploited in an efficient way only if the so-called affine dependence assumption on the operator and right-hand side of the considered problem with respect to the parameters is satisfied. When it is not, the Empirical Interpolation Method is usually used to recover this assumption approximately. In both cases, the Reduced Basis Method requires to access and modify the assembly routines of the corresponding computational code, leading to an intrusive procedure. In this work, we derive variants of the EIM algorithm and explain how they can be used to turn the Reduced Basis Method into a nonintrusive procedure. We present examples of aeroacoustic problems solved by integral equations and show how our algorithms can benefit from the linear algebra tools available in the considered code.

Keywords

Reduced basis method Nonintrusive approximation Empirical interpolation method Aeroacoustics 

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References

  1. 1.
    Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. 339(9), 667–672 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Carpentieri, B., Duff, I., Giraud, L., Sylvand, G.: Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations. SIAM J. Sci. Comput. 27(3), 774–792 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Casenave, F.: Reduced Order Methods Applied to Aeroacoustic Problems Solved by Integral Equations. PhD thesis, Université Paris-Est (2013)Google Scholar
  4. 4.
    Casenave, F., Ern, A., Lelivre, T.: Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method. ESAIM: Math. Model. Numer. Anal. 48, 207–229 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Casenave, F., Ern, A., Sylvand, G.: Coupled BEM-FEM for the convected Helmholtz equation with non-uniform flow in a bounded domain. J. Comput. Phys. 257(Part A), 627–644 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    DeVore, R.A., Petrova, G., Wojtaszczyk, P.: Greedy algorithms for reduced bases in Banach spaces. Constr. Approx. 37(3), 455–466 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Huynh, D.B.P., Patera, A.T., Rozza, G., Sen, S.: A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. 345(8), 473–478 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Oliveira, I.B., Patera, A.T., Machiels, L., Maday, Y., Rovas, D.V.: Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris, Ser. I 331(2), 153–158 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Lassila, T., Manzoni, A., Rozza, G.: On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition. ESAIM: Math. Model. Numer. Anal. 46, 1555–1576 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Machiels, L., Maday, Y., Patera, A.T., Prud’ homme, C., Rovas, D.V., Turinici, G., Veroy, K.: Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods. CJ Fluids Eng. 124, 70–80 (2002)CrossRefGoogle Scholar
  11. 11.
    Maday, Y., Nguyen, N.C., Patera, A.T., Pau, S.: A general multipurpose interpolation procedure: the magic points. Commun. Pur. Appl. Anal. 8(1), 383–404 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mises, R.V., Pollaczek-Geiringer, H.: Praktische verfahren der gleichungsaufläsung. ZAMM - J. Appl. Math. Mech. / Zeitschrift für Angewandte Mathematik und Mechanik 9(1), 58–77 (1929)zbMATHCrossRefGoogle Scholar
  13. 13.
    Patera, A. T., Prud’ homme, C., Rovas, D. V., Veroy, K.: A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: Proceedings of the 16th AIAA Computational Fluid Dynamics Conference (2003)Google Scholar
  14. 14.
    Sylvand, G.: La méthode multipôle rapide en électromagntisme: performances, parallélisation, applications. PhD thesis, Université de Nice-Sophia Antipolis (2002)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Fabien Casenave
    • 1
    Email author
  • Alexandre Ern
    • 1
  • Tony Lelièvre
    • 1
    • 2
  1. 1.Université Paris-Est, CERMICS (ENPC)Marne-la-Vallée Cedex 2France
  2. 2.INRIA Rocquencourt, MICMAC Team-Project, Domaine de VoluceauLe Chesnay CedexFrance

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