RBF collocation path-following approach: optimal choice for shape parameter based on genetic algorithm

This paper presents a new method to solve a challenging problem and a topic of current research namely the selection of optimal shape parameters for the Radial Basis Function (RBF) collocation methods in  both interpolation and nonlinear Partial Differential Equations (PDEs) problems.  To this intent, a compromise must be made to achieve the conflict between accuracy and stability referred to as the trade-off  or uncertainty principle.  The use of genetic algorithm and path-following continuation allows us on the one hand to avoid the local optimum issue associated with RBF interpolation matrices, which are inherently ill-conditioned and on the other side, to  map the original optimization problem of defining a shape parameter into a root-finding problem.  Our computational experiments applied on nonlinear problems in structural calculations using our proposed adaptive algorithm based on genetic optimization with automatic selection of the shape parameter can yield more accuracy and a good precision compared to the same state of the art algorithm from literature with a fixed and given shape parameter and Finite Element Method (FEM).

  1. Buhmann M.  Radial basis functions: theory and implementations.  Cambridge Monographs on Applied and Computational Mathematics.  Cambridge: Cambridge University Press (2003).
  2. Lucy L. B.  A numerical approach to the testing of the fission hypothesis.  Astronomical Journal. 82, 1013–1024 (1977).
  3. Gingold R. A., Monaghan J. J.  Smoothed particle hydrodynamics : theory and application to non-spherical stars.  Monthly Notices of the Royal Astronomical Society. 181, 375–389 (1977).
  4. Melenk J. M., Babuska I.  The partition of unity finite element method: Basic theory and applications.  Computer Methods in Applied Mechanics and Engineering. 139 (1–4), 289–314 (1996).
  5. Belytschko T., Lu Y.Y., Gu L.  Element free Galerkin Methods.  International Journal for Numerical Methods in Engineering. 37, 229–256 (1994).
  6. Kansa E. J.  Multiquadrics – a scattered data approximation scheme with applications to computational fluid dynamics – I surface approximations and partial derivative estimates.  Computers & Mathematics with Applications. 19 (8–9), 127–145 (1990).
  7. Timesli A., Braikat B., Lahmam H., Zahrouni H.  A new algorithm based on Moving Least Square method to simulate material mixing in friction stir welding.  Engineering Analysis with Boundary Elements. 50, 372–380 (2015).
  8. Timesli A.  Optimized radius of influence domain in meshless approach for modeling of large deformation problems.  Iranian Journal of Science and Technology-Transactions of Mechanical Engineering (2021).
  9. Nayroles B., Touzot G., Villon P.  Generalizing the finite element method : diffuse approximation and diffuse elements.  Computational Mechanics. 10, 307–318 (1992).
  10. 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.  Computers & Mathematics with Applications. 19 (8–9), 147–161 (1990).
  11. Hardy R. L.  Multiquadric equations of topography and other irregular surfaces.  Journal of Geophysical Research. 76 (8), 1905–1915 (1971).            
  12. Yoon J.  Spectral approximation orders of radial basis function interpolation on the Sobolev space.  SIAM Journal on Mathematical Analysis. 33 (4), 946–958 (2001).
  13. Madych W. R.  Miscellaneous error bounds for multiquadric and related interpolators.  Computers and Mathematics with Applications. 24 (12), 121–138 (1992).            
  14. Cheng A. H. D., Golberg M. A., Kansa E. J., Zammito G.  Exponential convergence and H-c multiquadric collocation method for partial differential equations.  Numerical Methods of Partial Differential Equations. 19, 571-594 (2003).
  15. Buhmann M., Dyn N.  Spectral convergence of multiquadric interpolation.  Proceedings of the Edinburgh Mathematical Society. 36 (2), 319–333 (1993).    
  16. Powell M. J. D.  The theory of radial basis function approximation in 1990.  Advances in numerical Analysis. 2, 105–209 (1990).
  17. Hon Y. C.  A quasi-radial basis functions method for American options pricing.  Computers & Mathematics with Applications. 43 (3–5), 513–524 (2002).
  18. Esmaeilbeigi M., Hosseini M. M., Syed Tauseef M. D.  A new approach of the radial basis functions method for telegraph equations.  International Journal of Physical Sciences. 6 (6), 1517–1527 (2011).
  19. Bhatia G. S., Arora G.  Radial Basis Function Methods for Solving Partial Differential Equations-A Review.  Indian Journal of Science and Technology. 9 (45), 1–18 (2016).
  20. Ferreira A. J. M.  Thick Composite Beam Analysis Using a Global Meshless Approximation Based on Radial Basis Functions.  Mechanics of Advanced Materials and Structures. 10 (3), 271–284 (2003).
  21. Ferreira A. J. M, Carrera E., Cinefra M., Roque C. M. C.  Analysis of laminated doubly-curved shells by a layerwise theory and radial basis functions collocation, accounting for through-the-thickness deformations.  Computational Mechanics. 48, 13–25 (2011).
  22. Chen C. S., Fan C. M, Wen P. H.  The method of particular solutions for solving elliptic problems with variable coefficients.  The International Journal for Numerical Methods in Biomedical Engineering. 8 (3), 545–559 (2011).
  23. Chen C. S., Fan C. M, Wen P. H.  The method of particular solutions for solving certain partial differential equations.  Numerical Methods for Partial Differential Equations. 28, 506–522 (2012).
  24. Fornberg B., Wright G.  Stable computation of multiquadric interpolants for all values of the shape parameter.  Computers & Mathematics with Applications. 48 (5–6), 853–867 (2004).
  25. Fornberg B., Zuev J.  The Runge phenomenon and spatially variable shape parameters in RBF interpolation.  Computers & Mathematics with Applications. 54 (3), 379–398 (2007).
  26. Larsson E., Fornberg B.  Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions.  Computers & Mathematics with Applications. 49 (1), 103–130 (2005).
  27. Carlson R., Foley T.  The parameter R2 in multiquadric interpolation.  Computers and Mathematics with Applications. 21 (9), 29–42 (1991).
  28. Madych W. R., Nelson S. A.  Multivariate Interpolation and Conditionally Positive Definite Functions. II. Mathematics of Computation. 54, 211–230 (1990).
  29. Madych W. R., Nelson S. A.  Multivariate interpolation and conditionally positive definite functions.  Approximation Theory and its Applications. 4, 77–89 (1988).
  30. Wendland H.  Scattered Data Approximation (Cambridge Monographs on Applied and Computational Mathematics).  Cambridge: Cambridge University Press (2004).
  31. Madych W. R.  Error estimates for interpolation by generalized splines.  Curves and Surfaces. 297–306 (1991).
  32. Kansa E. J., Carlson R. E.  Improved accuracy of multiquadric interpolation using variable shape parameters.  Computers & Mathematics with Applications. 24 (12), 99–120 (1992).
  33. Franke R.  Scattered data interpolation: tests of some methods.  Mathematics of Computation. 38 (157), 181–200 (1982).    
  34. Fasshauer G. E., McCourt M.  Stable evaluation of gaussian radial basis function interpolants.  SIAM Journal on Scientific Computing. 34 (2), A737–A762 (2012).        
  35. Foley T. A.  Near optimal parameter selection for multiquadric interpolation.  Journal of Applied Science and Computation. 1, 54–69 (1994).    
  36. Rippa S.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation.  Advances in Computational Mathematics. 11, 193–210 (1999).                
  37. Chen W., Hong Y., Lin J.  The sample solution approach for determination of the optimal shape parameter in the Multiquadric function of the Kansa method.  Computers & Mathematics with Applications. 75 (8), 2942–2954 (2018).            
  38. Zheng S., Feng R., Huang A.  The Optimal Shape Parameter for the Least Squares Approximation Based on the Radial Basis Function.  Mathematics. 8 (11), 1923 (2020).           
  39. Afiatdoust F., Esmaeilbeigi M.  Optimal variable shape parameters using genetic algorithm for radial basis function approximation.  Ain Shams Engineering Journal. 6 (2), 639–647 (2015).
  40. Esmaeilbeigi M., Hosseini M.  A new approach based on the genetic algorithm for finding a good shape parameter in solving partial differential equations by Kansa's method.  Applied Mathematics and Computation. 249, 419–428 (2014).
  41. Biazar J., Hosami M.  Selection of an Interval for Variable Shape Parameter in Approximation by Radial Basis Functions.  Advances in Numerical Analysis. 2016, Article ID: 1397849 (2016).
  42. Weikuan J., Dean Z., Tian S., Chunyang S., Chanli H., Yuyan Z.  A New Optimized GA-RBF Neural Network Algorithm.  Computational Intelligence and Neuroscience. 2014, Article ID: 982045 (2014).
  43. Saffah Z., Timesli A., Lahmam H., Azouani A., Amdi M.  New collocation path-following approach for the optimal shape parameter using Kernel method.  SN Applied Sciences. 3, 249 (2021).            
  44. Timesli A.  Buckling analysis of double walled carbon nanotubes embedded in Kerr elastic medium under axial compression using the nonlocal Donnell shell theory.  Advances in Nano Research. 9 (2), 69–82 (2020).    
  45. Timesli A.  An efficient approach for prediction of the nonlocal critical buckling load of double-walled carbon nanotubes using the nonlocal Donnell shell theory.  SN Applied Sciences. 2, Article number: 407 (2020).
  46. Cochelin B.  A path-following technique via an asymptotic-numerical method.  Computer and Structures. 53 (5), 1181–1192 (1994).        
  47. Mitchell M.  An Introduction to Genetic Algorithms.  Cambridge, MA: MIT Press (1996).
  48. Hassouna S., Timesli A.  Optimal variable support size for mesh-free approaches using genetic algorithm.  Mathematical Modeling and Computing. 8 (4), 678–690 (2021).
  49. Schaback R.  Error estimates and condition numbers for radial basis function interpolation.  Advances in Computational Mathematics. 3, 251–264 (1995).
Mathematical Modeling and Computing, Vol. 8, No. 4, pp. 770–782 (2021)