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Preprints, Working Papers, ... Year : 2023

Extensions of the empirical interpolation method to vector-valued functions

Abstract

In industrial Computer-Assisted Engineering, it is common to deal with vector fields or multiple field variables. In this paper, different vector-valued extensions of the Empirical Interpolation Method (EIM) are considered. EIM has been shown to be a valuable tool for dimensionality reduction, reduced-order modeling for nonlinear problems and/or synthesis of families of solutions for parametric problems. Besides already existing vector-valued extensions, a new vector-valued EIM-the so-called VEIM approach-allowing interpolation on all the vector components is proposed and analyzed in this paper. This involves vector-valued basis functions, same magic points shared by all the components and linear combination matrices rather than scalar coefficients. Coefficient matrices are determined under constraints of point-wise interpolation properties for all the components and exact reconstruction property for the snapshots selected during the greedy iterative process. For numerical experiments, various vector-valued approaches including VEIM are tested and compared on various one, two and three-dimensional problems. All methods return robustness, stability and rather good convergence properties as soon as the Kolmogorov width of the dataset is not too big. Depending of the use case, a suitable and convenient method can be chosen among the different vector-valued EIM candidates.
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Dates and versions

hal-04171324 , version 1 (26-07-2023)

Licence

Attribution - NonCommercial - NoDerivatives

Identifiers

  • HAL Id : hal-04171324 , version 1

Cite

Florian de Vuyst, Kalinja Naffer-Chevassier, Yohann Goardou. Extensions of the empirical interpolation method to vector-valued functions. 2023. ⟨hal-04171324⟩
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