Abstract
In this paper, we present the first reduced basis method well-suited for the collocation framework. Two fundamentally different algorithms are presented: the so-called Least Squares Reduced Collocation Method (LSRCM) and Empirical Reduced Collocation Method (ERCM). This work provides a reduced basis strategy to practitioners who prefer a collocation, rather than Galerkin, approach. Furthermore, the empirical reduced collocation method eliminates a potentially costly online procedure that is needed for non-affine problems with Galerkin approach. Numerical results demonstrate the high efficiency and accuracy of the reduced collocation methods, which match or exceed that of the traditional reduced basis method in the Galerkin framework.
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Acknowledgements
The authors would like to thank Professor Maday, Yvon from Paris VI University for helpful discussions that led to a deeper understanding of the strength of our proposed approach. They also wish to thank the anonymous referees for constructive criticism that led to an improved presentation of the material in this paper.
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The research of Y. Chen was partially supported by National Science Foundation grant DMS-1216928, and by UMass Dartmouth Chancellor’s Research Fund and Joseph P. Healey Endowment Grants.
The research of S. Gottlieb was partially supported by AFOSR grant FA9550-09-1-0208.
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Chen, Y., Gottlieb, S. Reduced Collocation Methods: Reduced Basis Methods in the Collocation Framework. J Sci Comput 55, 718–737 (2013). https://doi.org/10.1007/s10915-012-9654-z
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DOI: https://doi.org/10.1007/s10915-012-9654-z