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Math. Comput. Appl. 2018, 23(1), 8; https://doi.org/10.3390/mca23010008

An Algorithmic Comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a Nonlinear Thermal Problem

1
EMMA—Efficient Methods for Mechanical Analysis, Institute of Applied Mechanics (CE), University of Stuttgart, Pfaffenwaldring 7, 70569 Stuttgart, Germany
2
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
3
MAT-Centre des Matériaux, MINES ParisTech, PSL Research University, CNRS UMR 7633, BP 87, 91003 Evry, France
4
Graduate School of Computational Engineering, Technische Universität Darmstadt, Dolivostraße 15, 64293 Darmstadt, Germany
*
Author to whom correspondence should be addressed.
Received: 19 December 2017 / Revised: 2 February 2018 / Accepted: 7 February 2018 / Published: 13 February 2018
(This article belongs to the Section Engineering)
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Abstract

A novel algorithmic discussion of the methodological and numerical differences of competing parametric model reduction techniques for nonlinear problems is presented. First, the Galerkin reduced basis (RB) formulation is presented, which fails at providing significant gains with respect to the computational efficiency for nonlinear problems. Renowned methods for the reduction of the computing time of nonlinear reduced order models are the Hyper-Reduction and the (Discrete) Empirical Interpolation Method (EIM, DEIM). An algorithmic description and a methodological comparison of both methods are provided. The accuracy of the predictions of the hyper-reduced model and the (D)EIM in comparison to the Galerkin RB is investigated. All three approaches are applied to a simple uncertainty quantification of a planar nonlinear thermal conduction problem. The results are compared to computationally intense finite element simulations. View Full-Text
Keywords: model order reduction (MOR); reduced basis model order reduction (RB MOR); uncertainty quantification (UQ); (discrete) empirical interpolation method (EIM; DEIM); hyper-reduction (HR) model order reduction (MOR); reduced basis model order reduction (RB MOR); uncertainty quantification (UQ); (discrete) empirical interpolation method (EIM; DEIM); hyper-reduction (HR)
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Fritzen, F.; Haasdonk, B.; Ryckelynck, D.; Schöps, S. An Algorithmic Comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a Nonlinear Thermal Problem. Math. Comput. Appl. 2018, 23, 8.

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