Abstract
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.
Similar content being viewed by others
References
Atluri, S.N., Cho, J.Y., Kim, H.-G.: Analysis of thin beams, using the meshless local Petrov–Galerkin method, with generalized moving least squares interpolations. Comput. Mech. 24(5), 334–347 (1999)
Backus, G., Gilbert, F.: The resolving power of gross earth data. Geophys. J. R. Astronom. Soc. 16(2), 169–205 (1968)
Barnett, G.A.: A Robust RBF-FD Formulation Based on Polyharmonic Splines and Polynomials. University of Colorado, Boulder (2015). Ph.D. thesis
Bayona, V.: An insight into RBF-FD approximations augmented with polynomials. Comput. Math. Appl. 77(9), 2337–2353 (2019)
Bayona, V., Flyer, N., Fornberg, B.: On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries. J. Comput. Phys. 380, 378–399 (2019)
Bayona, V., Flyer, N., Fornberg, B., Barnett, G.A.: On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs. J. Comput. Phys. 332, 257–273 (2017)
Belytschko, T., Krongauz, Y., Fleming, M., Organ, D., Liu, W.K.S.: Smoothing and accelerated computations in the element free Galerkin method. J. Comput. Appl. Math. 74(1–2), 111–126 (1996)
Bos, L.P., Salkauskas, K.: Moving least-squares are Backus–Gilbert optimal. J. Approx. Theory 59(3), 267–275 (1989)
Curtis, P.C., et al.: \(n\)-parameter families and best approximation. Pac. J. Math. 9(4), 1013–1027 (1959)
Dilts, G.A.: Moving-least-squares-particle hydrodynamics—I. Consistency and stability. Int. J. Numer. Methods Eng. 44(8), 1115–1155 (1999)
Dilts, G.A.: Moving least-squares particle hydrodynamics II: conservation and boundaries. Int. J. Numer. Methods Eng. 48(10), 1503–1524 (2000)
Duchon, J.: Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces, Revue française d’automatique, informatique, recherche opérationnelle. Analyse Numérique 10(R3), 5–12 (1976)
Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces, constructive theory of functions of several variables, pp. 85–100. Springer, Berlin (1977)
Duchon, J.: Sur l’erreur d’interpolation des fonctions de plusieurs variables par les \(D^m\)-splines. RAIRO Anal. Numérique 12(4), 325–334 (1978)
Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences, vol. 6. World Scientific Publishers, Singapore (2007)
Fleishman, S., Cohen-Or, D., Silva, C.T.: Robust moving least-squares fitting with sharp features. ACM Trans. Graph. (TOG) 24(3), 544–552 (2005)
Flyer, N., Barnett, G.A., Wicker, L.J.: Enhancing finite differences with radial basis functions: experiments on the Navier–Stokes equations. J. Comput. Phys. 316, 39–62 (2016)
Flyer, N., Fornberg, B., Bayona, V., Barnett, G.A.: On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy. J. Comput. Phys. 321, 21–38 (2016)
Fornberg, B., Flyer, N.: A Primer on Radial Basis Functions with Applications to the Geosciences. SIAM, Philadelphia (2015)
Franke, R.: Scattered data interpolation: tests of some methods. Math. Comput. 38(157), 181–200 (1982)
Guennebaud, G., Gross, M.: Algebraic point set surfaces. ACM Trans. Graph. (TOG) 26, 23 (2007)
Hardy, R.L.: Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76(8), 1905–1915 (1971)
Kansa, E.J.: Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Comput. Math. Appl. 19(8–9), 127–145 (1990)
Kansa, E.J.: Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19(8–9), 147–161 (1990)
Lancaster, P.: Moving weighted least-squares methods. Polynomial Spline Approx. 49, 103 (1979)
Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comput. 37(155), 141–158 (1981)
Levin, D.: The approximation power of moving least-squares. Math. Comput. Am. Math. Soc. 67(224), 1517–1531 (1998)
Liew, K.M., Huang, Y.Q., Reddy, J.N.: Moving least squares differential quadrature method and its application to the analysis of shear deformable plates. Int. J. Numer. Methods Eng. 56(15), 2331–2351 (2003)
Liew, K.M., Huang, Y.Q., Reddy, J.N.: Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method. Comput. Methods Appl. Mech. Eng. 192(19), 2203–2222 (2003)
Mairhuber, J.C.: On Haar’s theorem concerning Chebychev approximation problems having unique solutions. Proc. Am. Math. Soc. 7(4), 609–615 (1956)
McLain, D.H.: Drawing contours from arbitrary data points. Comput. J. 17(4), 318–324 (1974)
Meinguet, J.: An intrinsic approach to multivariate spline interpolation at arbitrary points. Polynomial Spline Approx. 49, 163–190 (1979)
Meinguet, J.: Multivariate interpolation at arbitrary points made simple. Z. Angew. Math. und Physik ZAMP 30(2), 292–304 (1979)
Micchelli, C.A.: Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions, Approximation Theory and Spline Functions, pp. 143–145. Springer, Berlin (1984)
Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)
Schaefer, S., McPhail, T., Warren, J.: Image deformation using moving least squares. ACM Trans. Graph. (TOG) 25, 533–540 (2006)
Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM National Conference, ACM, pp. 517–524 (1968)
Shu, C., Ding, H., Yeo, K.S.: Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 192(7–8), 941–954 (2003)
Tiwari, S., Kuhnert, J.: Finite Pointset Method Based on the Projection Method for Simulations of the Incompressible Navier–Stokes Equations, Meshfree Methods for Partial Differential Equations, pp. 373–387. Springer, Berlin (2003)
Tolstykh, A.I., Shirobokov, D.A.: On using radial basis functions in a “finite difference mode” with applications to elasticity problems. Comput. Mech. 33(1), 68–79 (2003)
Wright, G.B., Fornberg, B.: Scattered node compact finite difference-type formulas generated from radial basis functions. J. Comput. Phys. 212(1), 99–123 (2006)
Acknowledgements
The author would like to thank Bengt Fornberg, who took the time to carefully review this manuscript and make useful comments. This work was supported by Spanish MECD Grant FIS2016-77892-R.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bayona, V. Comparison of Moving Least Squares and RBF+poly for Interpolation and Derivative Approximation. J Sci Comput 81, 486–512 (2019). https://doi.org/10.1007/s10915-019-01028-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-019-01028-8