Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation

Abstract : We propose a probabilistic way for reducing the cost of classical projection-based model order reduction methods for parameter-dependent linear equations. A classical reduced order model is here approximated from its random sketch, which is a set of low-dimensional random projections of the reduced approximation space and the manifolds of associated residuals. This approach exploits the fact that the manifolds of parameter-dependent matrices and vectors involved in the full order model are contained in low-dimensional spaces. We provide conditions on the dimension of the random sketch for the resulting reduced order model to be quasi-optimal with high probability. Our approach can be used for reducing both complexity and memory requirements. The provided algorithms are well suited for any modern computational environment. Major operations, except solving linear systems of equations, are embarrassingly parallel. Our version of proper orthogonal decomposition can be computed on multiple workstations with a communication cost independent of the dimension of the full order model. The reduced order model can even be constructed in a so-called streaming environment, i.e., under extreme memory constraints. In addition, we provide an efficient way for estimating the error of the reduced order model, which is not only more efficient than the classical approach but is also less sensitive to round-off errors. Finally, the methodology is validated on benchmark problems.
Type de document :
Pré-publication, Document de travail
minor changes of notations and interpretations (e.g., a sketch is called a \Theta-sketch of a red.. 2018
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https://hal.archives-ouvertes.fr/hal-01899836
Contributeur : Anthony Nouy <>
Soumis le : vendredi 19 octobre 2018 - 19:42:07
Dernière modification le : lundi 22 octobre 2018 - 11:46:17

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  • HAL Id : hal-01899836, version 1
  • ARXIV : 1803.02602

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Oleg Balabanov, Anthony Nouy. Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation. minor changes of notations and interpretations (e.g., a sketch is called a \Theta-sketch of a red.. 2018. 〈hal-01899836〉

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