Kinetic theory models involving the Fokker-Planck equation can be accurately discretized using a mesh support (finite elements, finite differences, finite volumes, spectral techniques, etc.). However, these techniques involve a high number of approximation functions. In the finite element framework, widely used in complex flow simulations, each approximation function is related to a node that defines the associated degree of freedom. When the model involves high dimensional spaces (including physical and conformation spaces and time), standard discretization techniques fail due to an excessive computation time required to perform accurate numerical simulations. One appealing strategy that allows circumventing this limitation is based on the use of reduced approximation basis within an adaptive procedure making use of an efficient separation of variables.
International audienceThis paper revisits a powerful discretization technique, the Proper Generalized Decomposition-PGD, illustrating its ability for solving highly multidimensional models. This technique operates by constructing a separated representation of the solution, such that the solution complexity scales linearly with the dimension of the space in which the model is defined, instead the exponentially-growing complexity characteristic of mesh based discretization strategies. The PGD makes possible the efficient solution of models defined in multidimensional spaces, as the ones encountered in quantum chemistry, kinetic theory description of complex fluids, genetics (chemical master equation), financial mathematics, aEuro broken vertical bar but also those, classically defined in the standard space and time, to which we can add new extra-coordinates (parametric models, aEuro broken vertical bar) opening numerous possibilities (optimization, inverse identification, real time simulations, aEuro broken vertical bar)
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