An Error Estimator for Separated Representations of Highly Multidimensional Models

Amine Ammar (1) , Francisco Chinesta (2) , Pedro Diez (3) , Antonio Huerta (3)

1 LR - Laboratoire de rhéologie
2 GeM - Institut de Recherche en Génie Civil et Mécanique
3 LaCàN - Laboratori de Càlcul Numèric (LACAN)

Fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics. Most of these models consist of a partial differential equation defined in a highly multidimensional domain (e.g. Schrodinger equation, Fokker-Planck equations among many others). The main challenge related to these models is their associated curse of dimensionality. We proposed in some of our former works a new strategy able to circumvent the curse of dimensionality based on the use of separated representations (also known as finite sums decomposition). This technique proceeds by computing at each iteration a new sum that consists of a product of functions each one defined in one of the model coordinates. The issue related to error estimation has never been addressed. This paper presents a first attempt on the accuracy evaluation of such a kind of discretization techniques.

Mots clés: Finite sums decomposition; Separated representations; Error estimation; Curse of dimensionality; High dimensional models

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

Amine Ammar, Francisco Chinesta, Pedro Diez, Antonio Huerta. An Error Estimator for Separated Representations of Highly Multidimensional Models. Computer Methods in Applied Mechanics and Engineering, 2010, 199 (25-28), pp.1872-1880. ⟨10.1016/j.cma.2010.02.012⟩. ⟨hal-01004991⟩