Advertisement

On a Goal-Oriented Version of the Proper Generalized Decomposition Method

  • Kenan Kergrene
  • Ludovic Chamoin
  • Marc Laforest
  • Serge PrudhommeEmail author
Article
First Online:

Abstract

In this paper, we introduce, analyze, and numerically illustrate a goal-oriented version of the Proper Generalized Decomposition method. The objective is to derive a reduced-order formulation such that the accuracy in given quantities of interest is increased when compared to a standard Proper Generalized Decomposition method. Traditional goal-oriented methods usually compute the solution of an adjoint problem following the calculation of the primal solution for error estimation and adaptation. In the present work, we propose to solve the adjoint problem first, based on a reduced approach, in order to extract estimates of the quantities of interest and use this information to constrain the reduced primal problem. The resulting reduced-order constrained solution is thus capable of delivering more accurate estimates of the quantities of interest. The performance of the proposed approach is illustrated on several numerical examples.

Keywords

Model reduction Quantities of interest Constrained problem Mixed formulation Penalization Lagrange multiplier 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

SP is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. He also acknowledges the support by KAUST under Award Number OCRF-2014-CRG3-2281. Moreover, the authors gratefully acknowledge Olivier Le Maître for fruitful discussions on the subject.

References

  1. 1.
    Alfaro, I., González, D., Zlotnik, S., Díez, P., Cueto, E., Chinesta, F.: An error estimator for real-time simulators based on model order reduction. Adv. Model. Simul. Eng. Sci. 2(1), 30 (2015)CrossRefGoogle Scholar
  2. 2.
    Almeida, J.P.: A basis for bounding the errors of proper generalised decomposition solutions in solid mechanics. Int. J. Numer. Methods Eng. 94(10), 961–984 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ammar, A., Chinesta, F., Díez, P., Huerta, A.: An error estimator for separated representations of highly multidimensional models. Comput. Methods Appl. Mech. Eng. 199(25–28), 1872–1880 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Amsallem, D., Farhat, C.: Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46(7), 1803–1813 (2008)CrossRefGoogle Scholar
  5. 5.
    Babuška, I., Miller, A.: The post-processing approach in the finite element method—Part 1: calculation of displacements, stresses and other higher derivatives of the displacements. Int. J. Numer. Methods Eng. 20(6), 1085–1109 (1984)CrossRefzbMATHGoogle Scholar
  6. 6.
    Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Billaud-Friess, M., Nouy, A., Zahm, O.: A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems. ESAIM Math. Modell. Numer. Anal. 48(6), 1777–1806 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bochev, P.B., Gunzburger, M.D.: Least-Squares Finite Element Methods, vol. 166. Springer Science & Business Media, New York (2009)zbMATHGoogle Scholar
  9. 9.
    Boyaval, S., Le Bris, C., Lelièvre, T., Maday, Y., Nguyen, N.C., Patera, A.T.: Reduced basis techniques for stochastic problems. Arch. Comput. Methods Eng. 17(4), 435–454 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bui-Than, T., Willcox, K., Ghattas, O., van Bloemen Waanders, B.: Goal-oriented, model-constrained optimization for reduction of large-scale systems. J. Comput. Phys. 224(2), 880–896 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carlberg, K., Farhat, C.: A low-cost, goal-oriented “compact proper orthogonal decomposition” basis for model reduction of static systems. Int. J. Numer. Methods Eng. 86, 381–402 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chamoin, L., Pled, F., Allier, P.-E., Ladevèze, P.: A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems. Comput. Methods Appl. Mech. Eng. 327, 118–146 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chaudhry, J.H., Cyr, E.C., Liu, K., Manteuffel, T.A., Olson, L.N., Tang, L.: Enhancing least-squares finite element methods through a quantity-of-interest. SIAM J. Numer. Anal. 52(6), 3085–3105 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chen, P., Quarteroni, A., Rozza, G.: A weighted reduced basis method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 51(6), 3163–3185 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chinesta, F., Keunings, R., Leygue, A.: The Proper Generalized Decomposition for Advanced Numerical Simulations. Springer International Publishing, New York (2014)CrossRefzbMATHGoogle Scholar
  16. 16.
    Gunzburger, M.D., Peterson, J.S., Shadid, J.N.: Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comput. Methods Appl. Mech. Eng. 196(4), 1030–1047 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Karush, W.: Minima of functions of several variables with inequalities as side constraints. Master’s thesis, Department of Mathematics, University of Chicago, Chicago, Illinois (1939)Google Scholar
  18. 18.
    Kergrene, K., Prudhomme, S., Chamoin, L., Laforest, M.: Approximation of constrained problems using the PGD method with application to pure Neumann problems. Comput. Methods Appl. Mech. Eng. 317, 507–525 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kergrene, K., Prudhomme, S., Chamoin, L., Laforest, M.: A new goal-oriented formulation of the finite element method. Comput. Methods Appl. Mech. Eng. 327, 256–276 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492. University of California Press, Berkeley (1951)Google Scholar
  21. 21.
    Ladevèze, P., Chamoin, L.: On the verification of model reduction methods based on the proper generalized decomposition. Comput. Methods Appl. Mech. Eng. 200(23–24), 2032–2047 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ladevèze, P., Chamoin, L.: Toward guaranteed PGD-reduced models. Bytes and Science, pp. 143–154. CIMNE, Barcelona (2013)Google Scholar
  23. 23.
    Nouy, A.: A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations. Comput. Methods Appl. Mech. Eng. 23–24(199), 1603–1626 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Oden, J.T., Prudhomme, S.: Goal-oriented error estimation and adaptivity for the finite element method. Comput. Math. Appl. 41(5–6), 735–756 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Prudhomme, S., Oden, J.T.: On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput. Methods Appl. Mech. Eng. 176(1–4), 313–331 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)CrossRefzbMATHGoogle Scholar
  28. 28.
    Venturi, L., Torlo, D., Ballarin, F., Rozza, G.: A weighted POD method for elliptic PDEs with random inputs. ArXiv e-prints (2018)Google Scholar
  29. 29.
    Venturi, L., Torlo, D., Ballarin, F., Rozza, G.: Weighted reduced order methods for parametrized partial differential equations with random inputs. ArXiv e-prints (2018)Google Scholar
  30. 30.
    Zlotnik, S., Díez, P., Gonzalez, D., Cueto, E., Huerta, A.: Effect of the separated approximation of input data in the accuracy of the resulting PGD solution. Adv. Model. Simul. Eng. Sci. 2(1), 1–14 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Kenan Kergrene
    • 1
  • Ludovic Chamoin
    • 2
  • Marc Laforest
    • 1
  • Serge Prudhomme
    • 1
    Email author
  1. 1.Département de mathématiques et de génie industrielPolytechnique MontréalMontréalCanada
  2. 2.LMT ENS Paris-Saclay/CNRS/Université Paris-SaclayCachanFrance

Personalised recommendations

Cite article

Buy options