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January  2009, 8(1): 383-404. doi: 10.3934/cpaa.2009.8.383

A general multipurpose interpolation procedure: the magic points

Yvon Maday 1, , Ngoc Cuong Nguyen 2, , Anthony T. Patera 3, and  S. H. Pau 4,

1. 

UPMC Univ Paris 06,UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, Division of Applied Mathematics, Brown University, Providence, RI, United States
2. 

Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge MA02139, United States
3. 

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge MA02139, United States
4. 

Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley CA94720, United States

Received  July 2008 Revised  September 2008 Published  October 2008

Lagrangian interpolation is a classical way to approximate general functions by finite sums of well chosen, pre-defined, linearly independent interpolating functions; it is much simpler to implement than determining the best fits with respect to some Banach (or even Hilbert) norms. In addition, only partial knowledge is required (here values on some set of points). The problem of defining the best sample of points is nevertheless rather complex and is in general open. In this paper we propose a way to derive such sets of points. We do not claim that the points resulting from the construction explained here are optimal in any sense. Nevertheless, the resulting interpolation method is proven to work under certain hypothesis, the process is very general and simple to implement, and compared to situations where the best behavior is known, it is relatively competitive.
Keywords: polynomial interpolation, Empirical interpolation, magic points.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Yvon Maday, Ngoc Cuong Nguyen, Anthony T. Patera, S. H. Pau. A general multipurpose interpolation procedure: the magic points. Communications on Pure & Applied Analysis, 2009, 8 (1) : 383-404. doi: 10.3934/cpaa.2009.8.383
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