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

Article dans une revue avec comité de lecture
Author
ccPRULIERE, Etienne
ccCHINESTA SORIA, Francisco
10921 Institut de Recherche en Génie Civil et Mécanique [GeM]
ccAMMAR, Amine
705 Laboratoire de rhéologie [LR]

URI
http://hdl.handle.net/10985/6274
DOI
10.1016/j.matcom.2010.07.015
Date
2010

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 encoun- tered 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.

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