Circumventing Curse of Dimensionality in the Solution of Highly Multidimensional Models Encountered in Quantum Mechanics Using Meshfree Finite Sums Decomposition

Amine Ammar (1, 2) , Francisco Chinesta (3)

1 LRP - Laboratoire Rhéologie et Procédés
2 LR - Laboratoire de rhéologie
3 LMSP - Laboratoire de Mécanique des Systèmes et des Procédés

The fine description of the mechanics and structure of materials at nanometric scale introduces some specific challenges related to the impressive number of degrees of freedom required due to the highly dimensional spaces in which those models are defined. This is the case of quantum mechanics models, in which the wavefunction is defined in a space of dimension 3×Np, being Np the number of particles involved, that leads to the terrific curse of dimensionality. Despite the fact that spectacular progresses have been accomplished in the context of computational mechanics in the last decade, the treatment of those models, as we describe in the present work, needs further developments.

Mots clés: Curse of dimensionality; Quantum Mechanics; Separated representations; Schrödinger equation

Domaines: Mécanique [physics.med-ph]

Amine Ammar, Francisco Chinesta. Circumventing Curse of Dimensionality in the Solution of Highly Multidimensional Models Encountered in Quantum Mechanics Using Meshfree Finite Sums Decomposition. Lecture Notes on Computational Science and Engineering, Springer, pp.1-17, 2008, Meshfree Methods for Partial Differential Equations IV, ⟨10.1007/978-3-540-79994-8_1⟩. ⟨hal-00289677⟩