An improved radial basis-pseudospectral method with hybrid Gaussian-cubic kernels

Abstract While pseudospectral (PS) methods can feature very high accuracy, they tend to be severely limited in terms of geometric flexibility. Application of global radial basis functions overcomes this, however at the expense of problematic conditioning (1) in their most accurate flat basis function regime, and (2) when problem sizes are scaled up to become of practical interest. The present study considers a strategy to improve on these two issues by means of using hybrid radial basis functions that combine cubic splines with Gaussian kernels. The parameters, controlling Gaussian and cubic kernels in the hybrid RBF, are selected using global particle swarm optimization. The proposed approach has been tested with radial basis-pseudospectral method for numerical approximation of Poisson, Helmholtz, and Transport equation. It was observed that the proposed approach significantly reduces the ill-conditioning problem in the RBF-PS method, at the same time, it preserves the stability and accuracy for very small shape parameters. The eigenvalue spectra of the coefficient matrices in the improved algorithm were found to be stable even at large degrees of freedom, which mimic those obtained in pseudospectral approach. Also, numerical experiments suggest that the hybrid kernel performs significantly better than both pure Gaussian and pure cubic kernels.

[1]  Manuel Kindelan,et al.  Radial basis function interpolation in the limit of increasingly flat basis functions , 2016, J. Comput. Phys..

[2]  Xiaoyong Liu,et al.  Radial Basis Function Neural Network Based on PSO with Mutation Operation to Solve Function Approximation Problem , 2010, ICSI.

[3]  S. Sarra,et al.  Regularized symmetric positive definite matrix factorizations for linear systems arising from RBF interpolation and differentiation , 2014 .

[4]  G. Wright,et al.  A hybrid radial basis function–pseudospectral method for thermal convection in a 3‐D spherical shell , 2010 .

[5]  Gregory E. Fasshauer,et al.  Analysis of natural frequencies of composite plates by an RBF-pseudospectral method , 2007 .

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

[7]  Scott A. Sarra,et al.  A random variable shape parameter strategy for radial basis function approximation methods , 2009 .

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

[9]  A. Cheng.  Multiquadric and its shape parameter—A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation , 2012 .

[10]  Manuel Kindelan,et al.  RBF-FD formulas and convergence properties , 2010, J. Comput. Phys..

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

[12]  Gregory E. Fasshauer,et al.  Preconditioning of Radial Basis Function Interpolation Systems via Accelerated Iterated Approximate Moving Least Squares Approximation , 2009 .

[13]  Michael J. McCourt,et al.  Stable Evaluation of Gaussian Radial Basis Function Interpolants , 2012, SIAM J. Sci. Comput..

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

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

[16]  Guirong Liu.  Mesh Free Methods: Moving Beyond the Finite Element Method , 2002 .

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

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

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

[20]  Elisabeth Larsson,et al.  Stable Computations with Gaussian Radial Basis Functions , 2011, SIAM J. Sci. Comput..

[21]  C.-S. Huang,et al.  On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs , 2010 .

[22]  A. Krowiak.  Radial basis function-based pseudospectral method for static analysis of thin plates , 2016 .

[23]  James Kennedy,et al.  Particle swarm optimization , 2002, Proceedings of ICNN'95 - International Conference on Neural Networks.

[24]  B. Walczak,et al.  Particle swarm optimization (PSO). A tutorial , 2015 .

[25]  Mrinal K. Sen,et al.  Hybrid Gaussian-cubic radial basis functions for scattered data interpolation , 2015, Computational Geosciences.

[26]  Gregory E. Fasshauer,et al.  Kernel-based Approximation Methods using MATLAB , 2015, Interdisciplinary Mathematical Sciences.

[27]  Bengt Fornberg,et al.  Stable computations with flat radial basis functions using vector-valued rational approximations , 2016, J. Comput. Phys..

[28]  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 .

[29]  Stefano De Marchi,et al.  A new stable basis for radial basis function interpolation , 2013, J. Comput. Appl. Math..

[30]  N. Mozayani,et al.  Adjusting the parameters of radial basis function networks using Particle Swarm Optimization , 2009, 2009 IEEE International Conference on Computational Intelligence for Measurement Systems and Applications.

[31]  C.M.C. Roque,et al.  Numerical experiments on optimal shape parameters for radial basis functions , 2009 .

[32]  G. Fasshauer.  RBF Collocation Methods As PseudospectralMethods , 2005 .

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

[34]  A. Cheng,et al.  Direct solution of ill‐posed boundary value problems by radial basis function collocation method , 2005 .

[35]  A. Biswas,et al.  Application of Global Particle Swarm Optimization for Inversion of Residual Gravity Anomalies Over Geological Bodies with Idealized Geometries , 2016, Natural Resources Research.

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

[37]  Kamran Behdinan,et al.  Particle swarm approach for structural design optimization , 2007 .

[38]  Erik Lehto,et al.  A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere , 2012, J. Comput. Phys..

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

[40]  E. J. Kansa,et al.  Multi-quadrics-a scattered data approximation scheme with applications to computational fluid dynamics-II , 1990 .

[41]  Y. Sanyasiraju,et al.  Local RBF‐FD solutions for steady convection–diffusion problems , 2007 .

[42]  Wen Chen,et al.  Recent Advances in Radial Basis Function Collocation Methods , 2013 .

[43]  Gregory E. Fasshauer,et al.  Computation of natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method , 2006 .

[44]  S. Sarra.  A numerical study of the accuracy and stability of symmetric and asymmetric RBF collocation methods for hyperbolic PDEs , 2008 .

[45]  Gregory E. Fasshauer,et al.  On choosing “optimal” shape parameters for RBF approximation , 2007, Numerical Algorithms.

[46]  R. Franke.  A Critical Comparison of Some Methods for Interpolation of Scattered Data , 1979 .

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

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

[49]  Marjan Uddin,et al.  RBF-PS scheme for solving the equal width equation , 2013, Appl. Math. Comput..

[50]  G. Fasshauer.  RBF Collocation Methods and Pseudospectral Methods , 2004 .

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

[52]  S. Sarra,et al.  A numerical study of a technique for shifting eigenvalues of radial basis function differentiation matrices , 2011 .