A Greedy Algorithm for Numerical Methods in High Dimension

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

1 LAMPA - Laboratoire Angevin de Mécanique, Procédés et InnovAtion
2 GeM - Institut de Recherche en Génie Civil et Mécanique
3 Departamento de Ciencias, Físicas, Matemáticas y de la Computación

In this work we study the problem of compute the best rank-r approximation to the solution of a class of linear systems. It arises in the discretized equations appearing in various physical domains, such as kinetic theory, statistical mechanics, quantum mechanics,and in nano-science and nanotechnology among others. In particular, we use the fact that tensors of order 3 or higher have best rank-1 approximation. Then we propose an iterative method such that at step-n we are to be able to compute an approximate solution of rank-n satisfying an optimal condition. Finally, we describe its relationship with the Finite Element Method for High-Dimensional Partial Differential Equations based on the tensorial product of one-dimensional bases. We illustrate this situation taking as a model problem the multidimensional Poisson equation with homogeneous Dirichlet boundary condition.

Mots clés: Finite Element Method; Rank-r approximation; Separated representation; Greedy Algorithm

Domaines: Mécanique [physics.med-ph]; Matériaux

Amine Ammar, Francisco Chinesta, Antonio Falcó. A Greedy Algorithm for Numerical Methods in High Dimension. International Workshop in Dynamical Systems and Multidisciplinary Applications, 2008, Elche, Spain. ⟨hal-01007749⟩