# American Institute of Mathematical Sciences

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

## A general multipurpose interpolation procedure: the magic points

 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.
Citation: Yvon Maday, Ngoc Cuong Nguyen, Anthony T. Patera, S. H. Pau. A general multipurpose interpolation procedure: the magic points. Communications on Pure and 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 and 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 and 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 and Continuous Dynamical Systems, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477 [4] Anh N. Le. Sublacunary sets and interpolation sets for nilsequences. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1855-1871. doi: 10.3934/dcds.2021175 [5] Jean Dolbeault, An Zhang. Parabolic methods for ultraspherical interpolation inequalities. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022080 [6] Anita Mayo. Accurate two and three dimensional interpolation for particle mesh calculations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1205-1228. doi: 10.3934/dcdsb.2012.17.1205 [7] V. Rehbock, K.L. Teo, L.S. Jennings. Suboptimal feedback control for a class of nonlinear systems using spline interpolation. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 223-236. doi: 10.3934/dcds.1995.1.223 [8] Rolando Mosquera, Aziz Hamdouni, Abdallah El Hamidi, Cyrille Allery. POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1743-1759. doi: 10.3934/dcdss.2019115 [9] Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 [10] Silvia Allavena, Michele Piana, Federico Benvenuto, Anna Maria Massone. An interpolation/extrapolation approach to X-ray imaging of solar flares. Inverse Problems and Imaging, 2012, 6 (2) : 147-162. doi: 10.3934/ipi.2012.6.147 [11] Lucio Boccardo, Daniela Giachetti. A nonlinear interpolation result with application to the summability of minima of some integral functionals. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 31-42. doi: 10.3934/dcdsb.2009.11.31 [12] Jeremy Levesley, Xinping Sun, Fahd Jarad, Alexander Kushpel. Interpolation of exponential-type functions on a uniform grid by shifts of a basis function. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2399-2416. doi: 10.3934/dcdss.2020403 [13] Antonella Falini, Francesca Mazzia, Cristiano Tamborrino. Spline based Hermite quasi-interpolation for univariate time series. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022039 [14] Gleb Beliakov. Construction of aggregation operators for automated decision making via optimal interpolation and global optimization. Journal of Industrial and Management Optimization, 2007, 3 (2) : 193-208. doi: 10.3934/jimo.2007.3.193 [15] Kohei Nakamura. An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1093-1102. doi: 10.3934/dcdss.2020385 [16] Matteo Bonforte, Jean Dolbeault, Bruno Nazaret, Nikita Simonov. Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022093 [17] Jean Dolbeault, Marta García-Huidobro, Rául Manásevich. Interpolation inequalities in $\mathrm W^{1,p}( {\mathbb S}^1)$ and carré du champ methods. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 375-394. doi: 10.3934/dcds.2020014 [18] Jinye Shen, Xian-Ming Gu. Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1179-1207. doi: 10.3934/dcdsb.2021086 [19] 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 and Continuous Dynamical Systems, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809 [20] François Golse, Clément Mouhot, Valeria Ricci. Empirical measures and Vlasov hierarchies. Kinetic and Related Models, 2013, 6 (4) : 919-943. doi: 10.3934/krm.2013.6.919

2021 Impact Factor: 1.273