is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. 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 meshbased 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, ... 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.

Models of kinetic theory provide a coarse-grained description of molecular configurations wherein atomistic processes are ignored. The Fokker-Planck equation related to the kinetic theory descriptions must be solved for the distribution function in both physical and configuration spaces. When the model involves high dimensional spaces (including physical and conformation spaces and time) standard discretization techniques fail due to excessive computation requirements. In this paper, we revisit some model reduction techniques recently proposed to circumvent those difficulties, exploring other new application areas related to entangled polymer models as well as the use of such reduced models for treating complex flows in which the distribution function involves both the physical and the conformation coordinates.