Hierarchical low-rank approximation of tensors by successive rank one corrections for preconditioning and solving high dimensional linear systems

Loïc Giraldi (1) , Anthony Nouy (1) , Grégory Legrain (1)

1 GeM - Institut de Recherche en Génie Civil et Mécanique

We propose an algorithm for preconditioning and solving high dimensional linear systems of equations in tensor format. The algorithm computes an approximation of a tensor in hierarchical Tucker format in a subspace constructed from successive rank one corrections. The algorithm can be applied for the approximation of a solution or of the inverse of an operator. In the latter case, properties such as sparsity or symmetry can be imposed to the approximation. The methodology is applied to high dimensional problems arising from the discretization of stochastic parametric problems.

Mots clés: high dimensional problems; low rank tensor approximation; hierarchical Tucker format; proper generalized decomposition; preconditioning

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

Loïc Giraldi, Anthony Nouy, Grégory Legrain. Hierarchical low-rank approximation of tensors by successive rank one corrections for preconditioning and solving high dimensional linear systems. 2nd ECCOMAS Young Investigators Conference (YIC 2013), Sep 2013, Bordeaux, France. ⟨hal-00855883⟩