Two and Three Dimensional Partition of Unity Interpolation by Product-Type Functions

In this paper we analyze the behavior of product-type radial basis functions (RBFs) and splines, which are used in a partition of unity interpolation scheme as local approximants. In particular, we deal with the case of bivariate and trivariate in terpolation on a relatively large number of scattered data points. Thus, we propose the local use of compactly supported RBF and spline interpolants, which take advantage of being expressible in the multivariate setting as a product of univariate functions. Numerical experiments show good accuracy and stability of the partition of unity method combined with these product-type interpolants, comparing it with the one obtained by replacing compactly supported RBFs and splines with Gaussians.

[1]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[2]  L. Montefusco,et al.  Radial basis functions for the multivariate interpolation of large scattered data sets , 2002 .

[3]  B. M. Fulk.  MATH , 1992 .

[4]  L. M. M.-T..  Theory of Probability , 1929, Nature.

[5]  Roberto Cavoretto,et al.  A class of spline functions for landmark‐based image registration , 2012 .

[6]  Roberto Cavoretto,et al.  Numerical integration on multivariate scattered data by Lobachevsky splines , 2013, Int. J. Comput. Math..

[7]  Robert J. Renka,et al.  Multivariate interpolation of large sets of scattered data , 1988, TOMS.

[8]  Martin D. Buhmann,et al.  Radial Basis Functions: Theory and Implementations: Preface , 2003 .

[9]  Jingwei Zhang,et al.  Algorithm 905 , 2010 .

[10]  Larry L. Schumaker,et al.  Approximation theory X : wavelets, splines, and applications , 2002 .

[11]  Layne T. Watson,et al.  SHEPPACK: a Fortran 95 package for interpolation using the modified Shepard algorithm , 2006, ACM-SE 44.

[12]  Bernhard Quatember,et al.  Radial Basis Functions and Splines for Landmark-Based Registration of Medical Images , 2010 .

[13]  Marc Duflot,et al.  Meshless methods: A review and computer implementation aspects , 2008, Math. Comput. Simul..

[14]  Andrew G. Glen,et al.  APPL , 2001 .

[15]  Eduardo Sáenz-de-Cabezón,et al.  A computer implementation of the partition of the unity procedure and its application to arthropod population dynamics. A case study on the European grape berry moth , 2011, Math. Comput. Simul..

[16]  P. Levy,et al.  Calcul des Probabilites , 1926, The Mathematical Gazette.

[17]  Roberto Cavoretto,et al.  A meshless interpolation algorithm using a cell-based searching procedure , 2014, Comput. Math. Appl..

[18]  Roberto Cavoretto.  A unified version of efficient partition of unity algorithms for meshless interpolation , 2012 .

[19]  Roberto Cavoretto,et al.  Lobachevsky spline functions and interpolation to scattered data , 2013 .

[20]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[21]  Roberto Cavoretto,et al.  Fast and accurate interpolation of large scattered data sets on the sphere , 2010, J. Comput. Appl. Math..

[22]  M. Urner.  Scattered Data Approximation , 2016 .

[23]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.

[24]  Roberto Cavoretto,et al.  Spherical interpolation using the partition of unity method: An efficient and flexible algorithm , 2012, Appl. Math. Lett..

[25]  Armin Iske,et al.  Armin Iske * Scattered Data Approximation by Positive Definite Kernel Functions , 2012 .

[26]  R. Brinks.  On the convergence of derivatives of B-splines to derivatives of the Gaussian function , 2008 .

[27]  Joachim Pouderoux,et al.  Adaptive hierarchical RBF interpolation for creating smooth digital elevation models , 2004, GIS '04.

[28]  Michael W. Berry,et al.  Algorithm 798: high-dimensional interpolation using the modified Shepard method , 1999, TOMS.

[29]  R. Cavoretto,et al.  Achieving accuracy and efficiency in spherical modelling of real data , 2014 .

[30]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[31]  Holger Wendland,et al.  Fast evaluation of radial basis functions : methods based on partition of unity , 2002 .

[32]  A. Rényi,et al.  Calcul des probabilités , 1966 .

[33]  Roberto Cavoretto,et al.  Multidimensional Lobachevsky Spline Integration on Scattered Data , 2014 .

[34]  Roberto Cavoretto,et al.  Scattered and track data interpolation using an efficient strip searching procedure , 2011, Appl. Math. Comput..

[35]  Roberto Cavoretto,et al.  A Trivariate Interpolation Algorithm Using a Cube-Partition Searching Procedure , 2014, SIAM J. Sci. Comput..

[36]  A. Borisenko.  Cylindrical multidimensional surfaces in Lobachevsky space , 1991 .