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On the deterministic solution of multidimensional parametric models using the Proper Generalized Decomposition

Author

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  • Pruliere, E.
  • Chinesta, F.
  • Ammar, A.

Abstract

This paper focuses on the efficient solution of models defined in high dimensional spaces. Those models involve numerous numerical challenges because of their associated curse of dimensionality. It is well known that in mesh-based discrete models the complexity (degrees of freedom) scales exponentially with the dimension of the space. Many models encountered in computational science and engineering involve numerous dimensions called configurational coordinates. Some examples are the models encountered in biology making use of the chemical master equation, quantum chemistry involving the solution of the Schrödinger or Dirac equations, kinetic theory descriptions of complex systems based on the solution of the so-called Fokker–Planck equation, stochastic models in which the random variables are included as new coordinates, financial mathematics, etc. This paper revisits the curse of dimensionality and proposes an efficient strategy for circumventing such challenging issue. This strategy, based on the use of a Proper Generalized Decomposition, is specially well suited to treat the multidimensional parametric equations.

Suggested Citation

  • Pruliere, E. & Chinesta, F. & Ammar, A., 2010. "On the deterministic solution of multidimensional parametric models using the Proper Generalized Decomposition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(4), pages 791-810.
    • Handle: RePEc:eee:matcom:v:81:y:2010:i:4:p:791-810
    DOI: 10.1016/j.matcom.2010.07.015
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    Cited by:

    1. Youngkyu Kim & Karen Wang & Youngsoo Choi, 2021. "Efficient Space–Time Reduced Order Model for Linear Dynamical Systems in Python Using Less than 120 Lines of Code," Mathematics, MDPI, vol. 9(14), pages 1-38, July.
    2. Oulghelou, M. & Allery, C., 2018. "A fast and robust sub-optimal control approach using reduced order model adaptation techniques," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 416-434.
    3. González, D. & Masson, F. & Poulhaon, F. & Leygue, A. & Cueto, E. & Chinesta, F., 2012. "Proper Generalized Decomposition based dynamic data driven inverse identification," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(9), pages 1677-1695.
    4. Metoui, S. & Pruliere, E. & Ammar, A. & Dau, F. & Iordanoff, I., 2018. "A multiscale separated representation to compute the mechanical behavior of composites with periodic microstructure," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 144(C), pages 162-181.
    5. Nicolas Courrier & Pierre-Alain Boucard & Bruno Soulier, 2016. "Variable-fidelity modeling of structural analysis of assemblies," Journal of Global Optimization, Springer, vol. 64(3), pages 577-613, March.
    6. Berger, Julien & Mendes, Nathan, 2017. "An innovative method for the design of high energy performance building envelopes," Applied Energy, Elsevier, vol. 190(C), pages 266-277.

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