Comparison of Moving Least Squares and RBF+poly for Interpolation and Derivative Approximation

The combination of polyharmonic splines (PHS) with high degree polynomials (PHS+poly) has recently opened new opportunities for radial basis function generated finite difference approximations. The PHS+poly formulation, which relies on a polynomial least squares fitting to enforce the local polynomial reproduction property, resembles somehow the so-called moving least squares (MLS) method. Although these two meshfree approaches are increasingly used nowadays, no direct comparison has been done yet. The present study aims to fill this gap, focusing on scattered data interpolation and derivative approximation. We first review the MLS approach and show that under some mild assumptions PHS+poly can be formulated analogously. Based on heuristic perspectives and numerical demonstrations, we then compare their performances in 1-D and 2-D. One key result is that, as previously found for PHS+poly, MLS can also overcome the edge oscillations (Runge’s phenomenon) by simply increasing the stencil size for a fixed polynomial degree. This is, however, controlled by a weighted least squares fitting which fails for high polynomial degrees. Overall, PHS+poly is found to perform superior in terms of accuracy and robustness.

[1]  Bengt Fornberg,et al.  On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs , 2017, J. Comput. Phys..

[2]  K. M. Liew,et al.  Moving least squares differential quadrature method and its application to the analysis of shear deformable plates , 2003 .

[3]  Bengt Fornberg,et al.  On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries , 2019, J. Comput. Phys..

[4]  G. Dilts.  MOVING-LEAST-SQUARES-PARTICLE HYDRODYNAMICS-I. CONSISTENCY AND STABILITY , 1999 .

[5]  Louis J. Wicker,et al.  Enhancing finite differences with radial basis functions: Experiments on the Navier-Stokes equations , 2016, J. Comput. Phys..

[6]  J. C. Mairhuber.  ON HAAR'S THEOREM CONCERNING CHEBYCHEV APPROXIMATION PROBLEMS HAVING UNIQUE SOLUTIONS' , 1956 .

[7]  G. Backus,et al.  The Resolving Power of Gross Earth Data , 1968 .

[8]  P. Lancaster.  Moving Weighted Least-Squares Methods , 1979 .

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

[10]  J. Meinguet.  Multivariate interpolation at arbitrary points made simple , 1979 .

[11]  M. Gross,et al.  Algebraic point set surfaces , 2007, SIGGRAPH 2007.

[12]  Jean Duchon,et al.  Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces , 1976 .

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

[14]  Hyun Gyu Kim,et al.  Analysis of thin beams, using the meshless local Petrov–Galerkin method, with generalized moving least squares interpolations , 1999 .

[15]  D. H. McLain,et al.  Drawing Contours from Arbitrary Data Points , 1974, Comput. J..

[16]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

[17]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[18]  K. M. Liew,et al.  Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method , 2003 .

[19]  K. Salkauskas,et al.  Moving least-squares are Backus-Gilbert optimal , 1989 .

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

[21]  D. Cohen-Or,et al.  Robust moving least-squares fitting with sharp features , 2005, ACM Trans. Graph..

[22]  J. Meinguet.  An Intrinsic Approach to Multivariate Spline Interpolation at Arbitrary Points , 1979 .

[23]  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 .

[24]  Bengt Fornberg,et al.  On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy , 2016, J. Comput. Phys..

[25]  C. Micchelli.  Interpolation of scattered data: Distance matrices and conditionally positive definite functions , 1986 .

[26]  David Levin,et al.  The approximation power of moving least-squares , 1998, Math. Comput..

[27]  J. Kuhnert,et al.  Finite Pointset Method Based on the Projection Method for Simulations of the Incompressible Navier-Stokes Equations , 2003 .

[28]  J. Duchon.  Sur l’erreur d’interpolation des fonctions de plusieurs variables par les $D^m$-splines , 1978 .

[29]  Mark A Fleming,et al.  Smoothing and accelerated computations in the element free Galerkin method , 1996 .

[30]  Gregory Barnett,et al.  A Robust RBF-FD Formulation based on Polyharmonic Splines and Polynomials , 2015 .

[31]  Jean Duchon,et al.  Splines minimizing rotation-invariant semi-norms in Sobolev spaces , 1976, Constructive Theory of Functions of Several Variables.

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

[33]  D. Shepard.  A two-dimensional interpolation function for irregularly-spaced data , 1968, ACM National Conference.

[34]  Victor Bayona,et al.  An insight into RBF-FD approximations augmented with polynomials , 2019, Comput. Math. Appl..

[35]  Philip C. Curtis.  $n$-parameter families and best approximation. , 1959 .

[36]  Bengt Fornberg,et al.  A primer on radial basis functions with applications to the geosciences , 2015, CBMS-NSF regional conference series in applied mathematics.

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

[38]  A. I. Tolstykh,et al.  On using radial basis functions in a “finite difference mode” with applications to elasticity problems , 2003 .

[39]  Gary A. Dilts,et al.  Moving least‐squares particle hydrodynamics II: conservation and boundaries , 2000 .

[40]  J. Warren,et al.  Image deformation using moving least squares , 2006, SIGGRAPH 2006.

[41]  M. Powell,et al.  Approximation theory and methods , 1984 .