Analytical and numerical advances in radial basis functions

Radial basis function (RBF) approximations have been used for some time to interpolate data on a sphere (as well as on many other types of domains). Their ability to solve, to spectral accuracy, convection-type PDEs over a sphere has been demonstrated only very recently. In such applications, there are two main choices that have to be made: (i) which type of radial function to use, and (ii) what value to choose for their shape parameter (denoted by ε, and with flat basis functions -stretched out in the radial directioncorresponding to ε = 0). The recent RBF-QR algorithm has made it practical to compute stably also for small values of ε. Results from solving a convective-type PDE on a sphere are compared here for many choices of radial functions over the complete range of ε-values (from very large down to the limit of ε → 0). The results are analyzed with a methodology that has similarities to the customary Fourier analysis in equispaced 1-D periodic settings. In particular, we find that high accuracy can be maintained also over very long time integrations. We furthermore gain insights into why RBFs sometimes offer higher accuracy than spherical harmonics (since the latter arise as an often non-optimal special case of the former). Anticipated future application areas for RBF-based methods in spherical geometries include weather and climate modeling.

[1]  Gabriele Steidl,et al.  Fast and stable algorithms for discrete spherical Fourier transforms , 1998 .

[2]  M. Buhmann.  On quasi-interpolation with radial basis functions , 1993 .

[3]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[4]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

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

[6]  Bengt Fornberg,et al.  On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere , 2008, J. Comput. Phys..

[7]  H. Kreiss,et al.  Comparison of accurate methods for the integration of hyperbolic equations , 1972 .

[8]  Stephen J. Thomas,et al.  The NCAR Spectral Element Climate Dynamical Core: Semi-Implicit Eulerian Formulation , 2005, J. Sci. Comput..

[9]  Paul N. Swarztrauber.  Spectral Transform Methods for Solving the Shallow-Water Equations on the Sphere , 1996 .

[10]  G. L. Browning,et al.  A comparison of three numerical methods for solving differential equations on the sphere , 1989 .

[11]  C. Shu,et al.  Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations , 2003 .

[12]  A. U.S.,et al.  Stable Computation of Multiquadric Interpolants for All Values of the Shape Parameter , 2003 .

[13]  R. Schaback,et al.  Characterization and construction of radial basis functions , 2001 .

[14]  J. Hack,et al.  Spectral transform solutions to the shallow water test set , 1995 .

[15]  Bengt Fornberg,et al.  Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids , 2005, Adv. Comput. Math..

[16]  B. Fornberg,et al.  The Gibbs Phenomenon for Radial Basis Functions , 2006 .

[17]  Bengt Fornberg,et al.  Comparison of finite difference‐ and pseudospectral methods for convective flow over a sphere , 1997 .

[18]  T. Driscoll,et al.  Interpolation in the limit of increasingly flat radial basis functions , 2002 .

[19]  B. Fornberg.  On a Fourier method for the integration of hyperbolic equations , 1975 .

[20]  Ian H. Sloan,et al.  How good can polynomial interpolation on the sphere be? , 2001, Adv. Comput. Math..

[21]  R. Schaback.  Multivariate Interpolation by Polynomials and Radial Basis Functions , 2005 .

[22]  B. Fornberg,et al.  Some observations regarding interpolants in the limit of flat radial basis functions , 2003 .

[23]  Bengt Fornberg,et al.  Scattered node compact finite difference-type formulas generated from radial basis functions , 2006, J. Comput. Phys..

[24]  Tilmann Gneiting,et al.  Normal scale mixtures and dual probability densities , 1997 .

[25]  Willi Freeden,et al.  Constructive Approximation on the Sphere: With Applications to Geomathematics , 1998 .

[26]  B. Fornberg,et al.  Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions , 2003 .

[27]  Mark Tygert,et al.  Fast Algorithms for Spherical Harmonic Expansions , 2006, SIAM J. Sci. Comput..

[28]  M. Powell,et al.  Radial basis function interpolation on an infinite regular grid , 1990 .

[29]  Philip E. Merilees,et al.  Numerical experiments with the pseudospectral method in spherical coordinates , 1974 .

[30]  Robin J. Y McLeod,et al.  Geometry and Interpolation of Curves and Surfaces , 1998 .

[31]  Natasha Flyer,et al.  Transport schemes on a sphere using radial basis functions , 2007, J. Comput. Phys..

[32]  John M. Dennis,et al.  A Comparison of Two Shallow-Water Models with Nonconforming Adaptive Grids , 2008 .

[33]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[34]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[35]  I. J. Schoenberg.  Metric spaces and completely monotone functions , 1938 .

[36]  Stephen J. Thomas,et al.  A Discontinuous Galerkin Transport Scheme on the Cubed Sphere , 2005 .

[37]  Jungho Yoon,et al.  Spectral Approximation Orders of Radial Basis Function Interpolation on the Sobolev Space , 2001, SIAM J. Math. Anal..

[38]  S. Orszag,et al.  Advanced Mathematical Methods For Scientists And Engineers , 1979 .

[39]  N SwarztrauberPaul,et al.  A standard test set for numerical approximations to the shallow water equations in spherical geometry , 1992 .

[40]  P. Swarztrauber,et al.  Fast Shallow-Water Equation Solvers in Latitude-Longitude Coordinates , 1998 .

[41]  Natasha Flyer,et al.  Exact polynomial reproduction for oscillatory radial basis functions on infinite lattices , 2006, Comput. Math. Appl..

[42]  W. Madych,et al.  Bounds on multivariate polynomials and exponential error estimates for multiquadratic interpolation , 1992 .

[43]  곽순섭,et al.  Generalized Functions , 2006, Theoretical and Mathematical Physics.

[44]  Elisabeth Larsson,et al.  A new class of oscillatory radial basis functions , 2006, Comput. Math. Appl..

[45]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

[46]  Francis X. Giraldo,et al.  A Scalable Spectral Element Eulerian Atmospheric Model (SEE-AM) for NWP: Dynamical Core Tests , 2004 .

[47]  D. Healy,et al.  Computing Fourier Transforms and Convolutions on the 2-Sphere , 1994 .

[48]  Philip E. Merilees,et al.  The pseudospectral approximation applied to the shallow water equations on a sphere , 1973 .

[49]  P. Swarztrauber,et al.  A standard test set for numerical approximations to the shallow water equations in spherical geometry , 1992 .

[50]  Nira Dyn,et al.  Spectral convergence of multiquadric interpolation , 1993, Proceedings of the Edinburgh Mathematical Society.

[51]  Clive Temperton.  On Scalar and Vector Transform Methods for Global Spectral Models , 1991 .

[52]  Andrew D. Back,et al.  Radial Basis Functions , 2001 .

[53]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[54]  B. Fornberg,et al.  A numerical study of some radial basis function based solution methods for elliptic PDEs , 2003 .

[55]  Ward Cheney,et al.  A course in approximation theory , 1999 .

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

[57]  Tobin A. Driscoll,et al.  Eigenvalue stability of radial basis function discretizations for time-dependent problems , 2006, Comput. Math. Appl..

[58]  E. Kansa,et al.  Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations , 2000 .

[59]  Martin J. Mohlenkamp.  A fast transform for spherical harmonics , 1997 .

[60]  Bengt Fornberg,et al.  The Runge phenomenon and spatially variable shape parameters in RBF interpolation , 2007, Comput. Math. Appl..

[61]  M. Taylor.  The Spectral Element Method for the Shallow Water Equations on the Sphere , 1997 .

[62]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[63]  R. E. Carlson,et al.  The parameter R2 in multiquadric interpolation , 1991 .

[64]  Bengt Fornberg,et al.  A Stable Algorithm for Flat Radial Basis Functions on a Sphere , 2007, SIAM J. Sci. Comput..

[65]  James J. Hack,et al.  Description of a Global Shallow Water Model Based on the Spectral Transform Method , 1992 .