HAL - INRIA :: [inria-00336911, version 1] Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations

inria-00336911, version 1

<?xml version="1.0" encoding="UTF-8"?>
<listBibl xmlns="http://www.tei-c.org/ns/1.0" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<listBibl type="article"><head>Documents sans référence de publication</head>
<biblStruct type="article" xml:id="inria-00336911"><analytic>
<author><persName><forename>Claude Le</forename><surname>Bris</surname></persName>
<affiliation>
<orgName type="laboratory">CERMICS</orgName>
<orgName type="team">Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique</orgName>
<country>FR</country>
<orgName type="institution">INRIA</orgName>
<orgName type="institution">Ecole Nationale des Ponts et Chaussées</orgName>
</affiliation>
<affiliation>
<orgName type="laboratory">INRIA Rocquencourt</orgName>
<orgName type="team">MICMAC</orgName>
<country>FR</country>
<orgName type="institution">INRIA</orgName>
<orgName type="institution">Ecole Nationale des Ponts et Chaussées</orgName>
</affiliation>
</author>
<author><persName><forename>Tony</forename><surname>Lelievre</surname></persName>
<affiliation>
<orgName type="laboratory">CERMICS</orgName>
<orgName type="team">Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique</orgName>
<country>FR</country>
<orgName type="institution">INRIA</orgName>
<orgName type="institution">Ecole Nationale des Ponts et Chaussées</orgName>
</affiliation>
<affiliation>
<orgName type="laboratory">INRIA Rocquencourt</orgName>
<orgName type="team">MICMAC</orgName>
<country>FR</country>
<orgName type="institution">INRIA</orgName>
<orgName type="institution">Ecole Nationale des Ponts et Chaussées</orgName>
</affiliation>
</author>
<author><persName><forename>Y.</forename><surname>Maday</surname></persName>
<affiliation>
<orgName type="laboratory">LJLL</orgName>
<orgName type="team">Laboratoire Jacques-Louis Lions</orgName>
<country>FR</country>
<orgName type="institution">CNRS : UMR7598</orgName>
<orgName type="institution">Université Pierre et Marie Curie - Paris VI</orgName>
</affiliation>
</author>
<title type="main" level="a">Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations</title></analytic><monogr><imprint/></monogr><idno type="HALid">http://hal.inria.fr/inria-00336911/en/</idno></biblStruct>
</listBibl>
</listBibl>

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

Claude Le Bris ^1, 2, Tony Lelievre (

) ^1, 2, Y. Maday ³

(04/11/2008)

Résumé : 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.

1 :	Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS)
	INRIA – Ecole Nationale des Ponts et Chaussées
2 :	MICMAC (INRIA Rocquencourt)
	INRIA – Ecole Nationale des Ponts et Chaussées
3 :	Laboratoire Jacques-Louis Lions (LJLL)
	CNRS : UMR7598 – Université Pierre et Marie Curie - Paris VI

Domaine

Mathématiques/Analyse numérique


inria-00336911, version 1
http://hal.inria.fr/inria-00336911/fr/
oai:hal.inria.fr:inria-00336911
Contributeur : Tony Lelievre <>
Soumis le : Mercredi 5 Novembre 2008, 14:58:41
Dernière modification le : Mercredi 5 Novembre 2008, 14:58:41