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

Published online by Cambridge University Press:  03 October 2014

M. Billaud-Friess ,
A. Nouy  and
O. Zahm
M. Billaud-Friess
Affiliation:
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
A. Nouy
Affiliation:
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
O. Zahm
Affiliation:
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
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Abstract

In this paper, we propose a method for the approximation of the solution ofhigh-dimensional weakly coercive problems formulated in tensor spaces using low-rankapproximation formats. The method can be seen as a perturbation of a minimal residualmethod with a measure of the residual corresponding to the error in a specified solutionnorm. The residual norm can be designed such that the resulting low-rank approximationsare optimal with respect to particular norms of interest, thus allowing to take intoaccount a particular objective in the definition of reduced order approximations ofhigh-dimensional problems. We introduce and analyze an iterative algorithm that is able toprovide an approximation of the optimal approximation of the solution in a given low-ranksubset, without any a priori information on this solution. We alsointroduce a weak greedy algorithm which uses this perturbed minimal residual method forthe computation of successive greedy corrections in small tensor subsets. We prove itsconvergence under some conditions on the parameters of the algorithm. The proposednumerical method is applied to the solution of a stochastic partial differential equationwhich is discretized using standard Galerkin methods in tensor product spaces.

Keywords

High-dimensional problemsnonlinear approximationlow-rank approximationproper generalized decompositionminimal residualstochastic partial differential equation

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Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 6 , November 2014 , pp. 1777 - 1806
Copyright
© EDP Sciences, SMAI 2014

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