Communications on Pure and Applied Analysis (CPAA)

A general multipurpose interpolation procedure: the magic points

Pages: 383 - 404, Volume 8, Issue 1, January 2009      doi:10.3934/cpaa.2009.8.383

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Yvon Maday - UPMC Univ Paris 06,UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, Division of Applied Mathematics, Brown University, Providence, RI, United States (email)
Ngoc Cuong Nguyen - Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge MA02139, United States (email)
Anthony T. Patera - Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge MA02139, United States (email)
S. H. Pau - Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley CA94720, United States (email)

Abstract: 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:  Empirical interpolation, polynomial interpolation, magic points
Mathematics Subject Classification:  Primary: 58F15, 58F17; Secondary: 53C35

Received: July 2008;      Revised: September 2008;      Available Online: October 2008.