Tensor representation of non-linear models using cross approximations

Jose Aguado (1) , Domenico Borzacchiello (1) , Kiran S Kollepara (1) , Francisco Chinesta (2) , Antonio Huerta (3)

1 ICI - Institut de Calcul Intensif
2 Arts et Métiers ParisTech
3 LaCàN - Laboratori de Càlcul Numèric (LACAN)

Tensor representations allow compact storage and efficient manipulation of multi-dimensional data. Based on these, tensor methods build low-rank subspaces for the solution of multi-dimensional and multi-parametric models. However, tensor methods cannot always be implemented efficiently , specially when dealing with non-linear models. In this paper, we discuss the importance of achieving a tensor representation of the model itself for the efficiency of tensor-based algorithms. We investigate the adequacy of interpolation rather than projection-based approaches as a means to enforce such tensor representation, and propose the use of cross approximations for models in moderate dimension. Finally, linearization of tensor problems is analyzed and several strategies for the tensor sub-space construction are proposed.

Mots clés: Multi-dimensional Problems; Non-linear Modeling; Low-rank Tensor Approximation; Proper Generalized Decomposition; Parametrized PDE; Cross Approximations; Reduced Order Modeling

Domaines: Analyse numérique [math.NA]; Sciences de l'ingénieur [physics]

Jose Aguado, Domenico Borzacchiello, Kiran S Kollepara, Francisco Chinesta, Antonio Huerta. Tensor representation of non-linear models using cross approximations. Journal of Scientific Computing, 2019, ⟨10.1007/s10915-019-00917-2⟩. ⟨hal-01996047v2⟩