A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems^∗

Ecole Centrale Nantes, Université de Nantes, GeM, UMR CNRS 6183, 1 rue de la Noë, BP 92101, 44321 Nantes Cedex 3, France.
marie.billaud-friess@ec-nantes.fr; anthony.nouy@ec-nantes.fr; olivier.zahm@ec-nantes.fr

Received: 14 March 2013
Revised: 21 March 2014

Abstract

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with a measure of the residual corresponding to the error in a specified solution norm. The residual norm can be designed such that the resulting low-rank approximations are optimal with respect to particular norms of interest, thus allowing to take into account a particular objective in the definition of reduced order approximations of high-dimensional problems. We introduce and analyze an iterative algorithm that is able to provide an approximation of the optimal approximation of the solution in a given low-rank subset, without any a priori information on this solution. We also introduce a weak greedy algorithm which uses this perturbed minimal residual method for the computation of successive greedy corrections in small tensor subsets. We prove its convergence under some conditions on the parameters of the algorithm. The proposed numerical method is applied to the solution of a stochastic partial differential equation which is discretized using standard Galerkin methods in tensor product spaces.