American Institute of Mathematical Sciences

Advanced
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
[1]

Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164
[2]

Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk, Michael Loss. Improved interpolation inequalities on the sphere. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695
[3]

Charles Fefferman. Interpolation by linear programming I. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477
[4]

Anita Mayo. Accurate two and three dimensional interpolation for particle mesh calculations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1205-1228. doi: 10.3934/dcdsb.2012.17.1205
[5]

V. Rehbock, K.L. Teo, L.S. Jennings. Suboptimal feedback control for a class of nonlinear systems using spline interpolation. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 223-236. doi: 10.3934/dcds.1995.1.223
[6]

Jeremy Levesley, Xinping Sun, Fahd Jarad, Alexander Kushpel. Interpolation of exponential-type functions on a uniform grid by shifts of a basis function. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020403
[7]

Rolando Mosquera, Aziz Hamdouni, Abdallah El Hamidi, Cyrille Allery. POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1743-1759. doi: 10.3934/dcdss.2019115
[8]

Silvia Allavena, Michele Piana, Federico Benvenuto, Anna Maria Massone. An interpolation/extrapolation approach to X-ray imaging of solar flares. Inverse Problems & Imaging, 2012, 6 (2) : 147-162. doi: 10.3934/ipi.2012.6.147
[9]

Lucio Boccardo, Daniela Giachetti. A nonlinear interpolation result with application to the summability of minima of some integral functionals. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 31-42. doi: 10.3934/dcdsb.2009.11.31
[10]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020250
[11]

Gleb Beliakov. Construction of aggregation operators for automated decision making via optimal interpolation and global optimization. Journal of Industrial & Management Optimization, 2007, 3 (2) : 193-208. doi: 10.3934/jimo.2007.3.193
[12]

Kohei Nakamura. An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020385
[13]

Jean Dolbeault, Marta García-Huidobro, Rául Manásevich. Interpolation inequalities in $ \mathrm W^{1,p}( {\mathbb S}^1) $ and carré du champ methods. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 375-394. doi: 10.3934/dcds.2020014
[14]

Jaume Llibre, Jesús S. Pérez del Río, J. Angel Rodríguez. Structural stability of planar semi-homogeneous polynomial vector fields applications to critical points and to infinity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809
[15]

François Golse, Clément Mouhot, Valeria Ricci. Empirical measures and Vlasov hierarchies. Kinetic & Related Models, 2013, 6 (4) : 919-943. doi: 10.3934/krm.2013.6.919
[16]

Jinying Ma, Honglei Xu. Empirical analysis and optimization of capital structure adjustment. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1037-1047. doi: 10.3934/jimo.2018191
[17]

Wu Chanti, Qiu Youzhen. A nonlinear empirical analysis on influence factor of circulation efficiency. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 929-940. doi: 10.3934/dcdss.2019062
[18]

Andreas Schadschneider, Armin Seyfried. Empirical results for pedestrian dynamics and their implications for modeling. Networks & Heterogeneous Media, 2011, 6 (3) : 545-560. doi: 10.3934/nhm.2011.6.545
[19]

Xueting Cui, Xiaoling Sun, Dan Sha. An empirical study on discrete optimization models for portfolio selection. Journal of Industrial & Management Optimization, 2009, 5 (1) : 33-46. doi: 10.3934/jimo.2009.5.33
[20]

Alireza Bahiraie, A.K.M. Azhar, Noor Akma Ibrahim. A new dynamic geometric approach for empirical analysis of financial ratios and bankruptcy. Journal of Industrial & Management Optimization, 2011, 7 (4) : 947-965. doi: 10.3934/jimo.2011.7.947

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (121)
  • HTML views (0)
  • Cited by (102)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences