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An instance optimality property for approximation problems with multiple approximation subspaces

Cedric Herzet 1 Mamadou Diallo 2 Patrick Héas 1 
1 SIMSMART - SIMulation pARTiculaire de Modèles Stochastiques
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
Abstract : Model-order reduction methods tackle the following general approximation problem: find an "easily-computable" but accurate approximationˆh approximationˆ approximationˆh of some target solution h. In order to achieve this goal, standard method-ologies combine two main ingredients: i) a set of problem-specific constraints; ii) some "simple" prior model on the set of target solutions. The most common prior model encountered in the literature assume that the target solution h is "close" to some low-dimensional subspace. Recently, triggered by the work by Binev et al. [5], several contributions have shown that refined prior models (based on a set of embedded approximation subspaces) may lead to enhanced approximation performance. Unfortunately, to date, no theoretical results have been derived to support the good empirical performance observed in these contributions. The goal of this work is to fill this gap. More specifically, we provide a mathematical characterization of the approximation performance achievable by some particular "multi-space" decoder and emphasize that, in some specific setups, this "multi-space" decoder has provably better recovery guarantees than its standard counterpart based on a single approximation subspace.
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Submitted on : Tuesday, November 6, 2018 - 11:20:00 AM
Last modification on : Friday, May 20, 2022 - 9:04:52 AM
Long-term archiving on: : Thursday, February 7, 2019 - 2:24:56 PM


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  • HAL Id : hal-01913339, version 1


Cedric Herzet, Mamadou Diallo, Patrick Héas. An instance optimality property for approximation problems with multiple approximation subspaces. 2018. ⟨hal-01913339⟩



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