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Contributions aux méthodes de calcul basées sur l'approximation de tenseurs et applications en mécanique numérique

Abstract : This thesis makes several contributions to the resolution of high dimensional problems in scientific computing, particularly for uncertainty quantification. We consider here variational problems formulated on tensor product spaces. We first propose an efficient preconditioning strategy for linear systems solved by iterative solvers using low-rank tensor approximations. The preconditioner is found as a low rank approximation of the inverse of the operator. A greedy algorithm allows to compute this approximation, possibly imposing symmetry or sparsity properties. This preconditioner is validated on symmetric and non-symmetric linear problems. We also make contributions to direct tensor approximation methods which consist in computing the best approximation of the solution of an equation in a set of low-rank tensors. These techniques, sometimes coined "Proper Generalized Decomposition" (PGD), define optimality with respect to suitable norms allowing an a priori approximation of the solution. In particular, we extend the classically used greedy algorithms to the sets of Tucker tensors and hierarchical Tucker tensors. To do so, we construct successive rank one corrections and update the approximation in the previous sets. The proposed algorithm can be interpreted as a construction of an increasing sequence of reduced spaces in which we compute a possibly approximate projection of the solution. The application of these methods to symmetric and non-symmetric problems shows the efficiency of this algorithm. The proposed preconditioner is also applied and allows to define a better norm for the approximation of the solution. We finally apply these methods to the numerical homogenization of heterogeneous materials whose geometry is extracted from images. We first present particular treatment of the geometry and the boundary conditions to use them in the tensor approximation framework. An adaptive approximation procedure based on an a posteriori error estimator is proposed to ensure a given accuracy on the quantities of interest which are the effective properties. The method is first developed to estimate effective thermal properties, and is extended to linear elasticity.
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Submitted on : Saturday, September 14, 2013 - 7:24:11 PM
Last modification on : Saturday, November 26, 2022 - 5:14:06 PM
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  • HAL Id : tel-00861986, version 1

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Loïc Giraldi. Contributions aux méthodes de calcul basées sur l'approximation de tenseurs et applications en mécanique numérique. Analyse numérique [math.NA]. Ecole Centrale de Nantes (ECN), 2012. Français. ⟨NNT : ⟩. ⟨tel-00861986⟩

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