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

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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.

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References

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Amsallem, D., Farhat, C.: Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46(7), 1803–1813 (2008)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  6. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bochev, P.B., Gunzburger, M.D.: Least-Squares Finite Element Methods, vol. 166. Springer Science & Business Media, New York (2009)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chinesta, F., Keunings, R., Leygue, A.: The Proper Generalized Decomposition for Advanced Numerical Simulations. Springer International Publishing, New York (2014)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ladevèze, P., Chamoin, L.: Toward guaranteed PGD-reduced models. Bytes and Science, pp. 143–154. CIMNE, Barcelona (2013)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  28. Venturi, L., Torlo, D., Ballarin, F., Rozza, G.: A weighted POD method for elliptic PDEs with random inputs. ArXiv e-prints (2018)

  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)

  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)

    Article  Google Scholar 

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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.

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Authors and Affiliations

  1. Département de mathématiques et de génie industriel, Polytechnique Montréal, Montréal, QC, H3T 1J4, Canada

    Kenan Kergrene, Marc Laforest & Serge Prudhomme

  2. LMT, ENS Paris-Saclay/CNRS/Université Paris-Saclay, 61 Avenue du Président Wilson, 94230, Cachan, France

    Ludovic Chamoin

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  1. Kenan Kergrene
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  2. Ludovic Chamoin
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  3. Marc Laforest
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  4. Serge Prudhomme
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Correspondence to Serge Prudhomme.

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Kergrene, K., Chamoin, L., Laforest, M. et al. On a Goal-Oriented Version of the Proper Generalized Decomposition Method. J Sci Comput 81, 92–111 (2019). https://doi.org/10.1007/s10915-019-00918-1

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