Approximation BKW

Résumé

En physique, l'approximation BKW (en l'honneur de Léon Brillouin, Hendrik Anthony Kramers et Gregor Wentzel) est une méthode développée en 1926 qui permet d'étudier le régime semi-classique d'un système quantique. La fonction d'onde est développée asymptotiquement au premier ordre de la puissance du quantum d'action \hbar . L'idée de base de la méthode BKW est que l'équation de Schrödinger se dérive de l'équation de propagation des ondes. On doit donc retrouver la mécanique classique dans la limite \hbar \rightarrow 0 comme on retrouve l'optique géométrique lorsque la longueur d'onde \lambda \rightarrow 0 dans la théorie de l'optique ondulatoire. L'approximation BKW (pour les francophones européens) est également connue sous les initiales WKB (pour les anglophones et les francophones nord-américains), WKBJ, BWKJ et parfois WBK ou BWK. Le J supplémentaire est pour le mathématicien Harold Jeffreys, qui a développé en 1923 une méthode généra

Source officielle

https://fr.wikipedia.org/wiki/Approximation_BKW

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In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a deterministic parameter-dependent problem on a high-dimensional parameter domain. Given a hierarchy of finite element discretizations for the spatial approximation, we make use of a multilevel framework in which we consider the differences of the solution on two consecutive finite element levels at the collocation points. We then address the approximation of these high-dimensional differences by adaptive low-rank tensor techniques. This allows to equilibrate the error on all levels by exploiting regularity and additional low-rank structure of the solution. We arrive at an explicit representation in a low-rank tensor format of the approximate solution on the entire parameter domain, which can be used for, e.g., the direct and cheap computation of statistics. Numerical results are provided in order to illustrate the approach.

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MATHICSE Technical Report : Multilevel tensor approximation of PDEs with random data

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