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On the computation of Proper Generalized Decomposition modes of parametric elliptic problems

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In a recent paper (Azaïez et al. in SIAM J Math Anal 50(5):5426–5445, https://doi.org/10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution we prove the linear convergence of the Power Iterate method applied to compute the modes of the PGD expansion, for both symmetric and non-symmetric problems, when the data are small. We also find a spectral convergence ratio of the PGD expansion in the mean parametric norm, for meaningful parametric elliptic problems.

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  1. 1.

    Ammar, A., Mokdad, B., Chinesta, F., Keunings, R.: A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J. Non Newton. Fluid Mech. 139, 153–176 (2006)

  2. 2.

    Ammar, A., Chinesta, F., Falcó, A.: On the convergence of a Greedy rank-one update algorithm for a class of linear systems. Arch. Comput. Methods Eng. 17(4), 473–486 (2010)

  3. 3.

    Azaïez, M., Ben-Belgacem, F., Casado-Díaz, J., Rebollo, T.Chacón, Murat, F.: A new algorithm of proper generalized decomposition for parametric symmetric elliptic problems. SIAM J. Math. Anal. 50(5), 5426–5445 (2018). https://doi.org/10.1137/17m1137164

  4. 4.

    Cancès, E., Lelievre, T., Ehrlacher, V.: Convergence of a greedy algorithm for high-dimensional convex nonlinear problems. Math Models Methods Appl Sci. 21(12), 2433–2467 (2011)

  5. 5.

    Chinesta, F., Ammar, A., Cueto, E.: Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models. Arch. Comput. Methods Eng. 17(4), 327–350 (2010)

  6. 6.

    Falcó, A., Nouy, A.: A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart Young approach. J. Math. Anal. Appl. 376, 469–480 (2011)

  7. 7.

    Falcó, A., Nouy, A.: Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces. Numer. Math. 121, 503–530 (2012)

  8. 8.

    Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)

  9. 9.

    Ladévèze, P.: Nonlinear Computational Structural Mechanics—New Approaches and Non-incremental Methods of Calculation. Springer, Berlin (1999)

  10. 10.

    Le Bris, C., Lelievre, T., Maday, Y.: Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations. Constr. Approx. 30(3), 621–651 (2009)

  11. 11.

    Nouy, A.: A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput. Methods Appl. Mech. Eng. 199(23–24), 1603–1626 (2010)

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This work has been partially supported by the Spanish Government - Feder EU grant MTM2015-64577-C2-1-R and RTI2018-093521-B-C31. The work of M. Azaïez has been partially supported by the Visiting Professors Program of the University of Sevilla (VIPPIT-2017-II.8).

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Correspondence to T. Chacón Rebollo.

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Azaïez, M., Chacón Rebollo, T. & Gómez Mármol, M. On the computation of Proper Generalized Decomposition modes of parametric elliptic problems. SeMA 77, 59–72 (2020). https://doi.org/10.1007/s40324-019-00198-7

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  • Proper Generalized Decomposition
  • Computation of PGD modes
  • Convergence rate analysis

Mathematics Subject Classification

  • 65N12
  • 65N99