Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations

Abstract : We investigate mathematically a nonlinear approximation type approach recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to solve high dimensional partial differential equations. We show the link between the approach and the greedy algorithms of approximation theory studied e.g. in [R.A. DeVore and V.N. Temlyakov, Adv. Comput. Math., 1996]. On the prototypical case of the Poisson equation, we show that a variational version of the approach, based on minimization of energies, converges. On the other hand, we show various theoretical and numerical difficulties arising with the non variational version of the approach, consisting of simply solving the first order optimality equations of the problem. Several unsolved issues are indicated in order to motivate further research.
Type de document :
Pré-publication, Document de travail
2008
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https://hal.inria.fr/inria-00336911
Contributeur : Tony Lelievre <>
Soumis le : mercredi 5 novembre 2008 - 14:58:41
Dernière modification le : vendredi 25 mai 2018 - 12:02:03

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  • HAL Id : inria-00336911, version 1
  • ARXIV : 0811.0474

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Claude Le Bris, Tony Lelievre, Yvon Maday. Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations. 2008. 〈inria-00336911〉

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