A priori tensor approximations for the numerical solution of high dimensional problems: alternative definitions

Gaël Bonithon (1) , Anthony Nouy (1)

1 GeM - Institut de Recherche en Génie Civil et Mécanique

Tensor-based methods are receiving a growing attention in computational science and engineering for the numerical solution of problems defined in high dimensional tensor spaces. A family of methods called Proper Generalized Decomposition (PGD) methods have been recently proposed. They introduce alternative definitions of tensor approximations, not based on natural best approximation problems, for the approximation to be computable without a priori information on the solution of problems. Here, we introduce and compare different definitions of PGD for the construction of tensor approximations. Convergence results are provided for some classes of variational problems and some variants of PGD. We also present how the PGD can be judiciously coupled with classical iterative methods where it is used as a solver of successive linear problems, thus allowing the use of a wider class of iterative methods compared to other tensor-based methods.

Mots clés: High-dimensional problems; Tensor approximation; Proper Generalized Decomposition; Greedy Algorithms; Model Reduction

Domaines: Analyse numérique [math.NA]

Gaël Bonithon, Anthony Nouy. A priori tensor approximations for the numerical solution of high dimensional problems: alternative definitions. 28th GAMM-Seminar Leipzig on Analysis and Numerical Methods in Higher Dimensions, Jan 2012, Leipzig, Germany. ⟨hal-00664061v1⟩