Revista de la
Unión Matemática Argentina
Stabilizing radial basis functions techniques for a local boundary integral method
Luciano Ponzellini Marinelli
Volume 64, no. 2 (2023), pp. 375–396    

https://doi.org/10.33044/revuma.2901

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Abstract

Radial basis functions (RBFs) have been gaining popularity recently in the development of methods for solving partial differential equations (PDEs) numerically. These functions have become an extremely effective tool for interpolation on scattered node sets in several dimensions. One key issue with infinitely smooth RBFs is the choice of a suitable value for the shape parameter $\varepsilon$, which controls the flatness of the function. It is observed that best accuracy is often achieved when $\varepsilon$ tends to zero. However, the system of discrete equations from interpolation matrices becomes ill-conditioned. A few numerical algorithms have been presented that are able to stably compute an interpolant, even in the increasingly flat basis function limit, such as the RBF-QR method and the RBF-GA method. We present these techniques in the context of boundary integral methods to improve the solution of PDEs with RBFs. These stable calculations open up new opportunities for applications and developments of local integral methods based on local RBF approximations. Numerical results for a small shape parameter that stabilizes the error are presented. Accuracy and comparisons are also shown for elliptic PDEs.

References

  1. M. Ahmad, Siraj-ul-Islam, and E. Larsson, Local meshless methods for second order elliptic interface problems with sharp corners, J. Comput. Phys. 416 (2020), 109500, 17 pp. MR 4094820.
  2. P. R. S. Antunes, Reducing the ill conditioning in the method of fundamental solutions, Adv. Comput. Math. 44 (2018), no. 1, 351–365. MR 3755753.
  3. V. Bayona, N. Flyer, B. Fornberg, and G. A. Barnett, On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs, J. Comput. Phys. 332 (2017), 257–273. MR 3591180.
  4. N. Caruso, M. Portapila, and H. Power, An efficient and accurate implementation of the localized regular dual reciprocity method, Comput. Math. Appl. 69 (2015), no. 11, 1342–1366. MR 3339046.
  5. D. A. Castro, W. F. Florez, M. Portapila, N. Caruso, C. A. Bustamante, R. Posada, and J. M. Granados, Numerical examination of the effect of different boundary conditions on the method of approximate particular solutions for scalar and vector problems, Eng. Anal. Bound. Elem. 127 (2021), 75–90. MR 4239676.
  6. M. Dehghan and M. Najafi, Numerical solution of a non-classical two-phase Stefan problem via radial basis function (RBF) collocation methods, Eng. Anal. Bound. Elem. 72 (2016), 111–127. MR 3550377.
  7. K. P. Drake and G. B. Wright, A stable algorithm for divergence-free radial basis functions in the flat limit, J. Comput. Phys. 417 (2020), 109595, 9 pp. MR 4106713.
  8. G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences, 6, World Scientific, Hackensack, NJ, 2007. MR 2357267.
  9. G. E. Fasshauer and M. J. McCourt, Stable evaluation of Gaussian radial basis function interpolants, SIAM J. Sci. Comput. 34 (2012), no. 2, A737–A762. MR 2914302.
  10. G. E. Fasshauer and M. J. McCourt, Kernel-Based Approximation Methods Using MATLAB, World Scientific, Hackensack, NJ, 2015. https://doi.org/10.1142/9335.
  11. N. Flyer, G. A. Barnett, and L. J. Wicker, Enhancing finite differences with radial basis functions: experiments on the Navier-Stokes equations, J. Comput. Phys. 316 (2016), 39–62. MR 3494343.
  12. N. Flyer and E. Lehto, Rotational transport on a sphere: local node refinement with radial basis functions, J. Comput. Phys. 229 (2010), no. 6, 1954–1969. MR 2586231.
  13. B. Fornberg and N. Flyer, A Primer on Radial Basis Functions with Applications to the Geosciences, CBMS-NSF Regional Conference Series in Applied Mathematics, 87, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2015. MR 3450067.
  14. B. Fornberg and N. Flyer, Fast generation of 2-D node distributions for mesh-free PDE discretizations, Comput. Math. Appl. 69 (2015), no. 7, 531–544. MR 3320269.
  15. B. Fornberg, E. Larsson, and N. Flyer, Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput. 33 (2011), no. 2, 869–892. MR 2801193.
  16. B. Fornberg and E. Lehto, Stabilization of RBF-generated finite difference methods for convective PDEs, J. Comput. Phys. 230 (2011), no. 6, 2270–2285. MR 2764546.
  17. B. Fornberg, E. Lehto, and C. Powell, Stable calculation of Gaussian-based RBF-FD stencils, Comput. Math. Appl. 65 (2013), no. 4, 627–637. MR 3011446.
  18. B. Fornberg and C. Piret, A stable algorithm for flat radial basis functions on a sphere, SIAM J. Sci. Comput. 30 (2007/08), no. 1, 60–80. MR 2377431.
  19. B. Fornberg and C. Piret, On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere, J. Comput. Phys. 227 (2008), no. 5, 2758–2780. MR 2388503.
  20. B. Fornberg, G. Wright, and E. Larsson, Some observations regarding interpolants in the limit of flat radial basis functions, Comput. Math. Appl. 47 (2004), no. 1, 37–55. MR 2062724.
  21. B. Fornberg and G. Wright, Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl. 48 (2004), no. 5-6, 853–867. MR 2105258.
  22. R. Franke, Scattered data interpolation: tests of some methods, Math. Comp. 38 (1982), no. 157, 181–200. MR 0637296.
  23. R. L. Hardy, Theory and applications of the multiquadric-biharmonic method. 20 years of discovery 1968–1988, Comput. Math. Appl. 19 (1990), no. 8-9, 163–208. MR 1040159.
  24. E. J. Kansa, Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates, Comput. Math. Appl. 19 (1990), no. 8-9, 127–145. MR 1040157.
  25. E. J. Kansa, 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 (1990), no. 8-9, 147–161. MR 1040158.
  26. E. Larsson and B. Fornberg, Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions, Comput. Math. Appl. 49 (2005), no. 1, 103–130. MR 2123189.
  27. C. A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx. 2 (1986), no. 1, 11–22. MR 0891767.
  28. P. K. Mishra, G. E. Fasshauer, M. K. Sen, and L. Ling, A stabilized radial basis-finite difference (RBF-FD) method with hybrid kernels, Comput. Math. Appl. 77 (2019), no. 9, 2354–2368. MR 3945118.
  29. E. H. Ooi and V. Popov, An efficient implementation of the radial basis integral equation method, Eng. Anal. Bound. Elem. 36 (2012), no. 5, 716–726. MR 2880719.
  30. P. W. Partridge, Radial basis approximation functions in the boundary element dual reciprocity method, Trans. Model. Simul. 23 (1999), 325–334.
  31. L. Ponzellini Marinelli, N. Caruso, and M. Portapila, A stable computation on local boundary-domain integral method for elliptic PDEs, Math. Comput. Simulation 180 (2021), 379–400. MR 4152797.
  32. V. Popov and T. Thanh Bui, A meshless solution to two-dimensional convection-diffusion problems, Eng. Anal. Bound. Elem. 34 (2010), no. 7, 680–689. MR 2639905.
  33. M. Portapila and H. Power, Iterative schemes for the solution of system of equations arising from the DRM in multi domain approach, and a comparative analysis of the performance of two different radial basis functions used in the interpolation, Eng. Anal. Bound. Elem. 29 (2005), no. 2, 107–125. https://doi.org/10.1016/j.enganabound.2004.08.008.
  34. S. A. Sarra, Integrated multiquadric radial basis function approximation methods, Comput. Math. Appl. 51 (2006), no. 8, 1283–1296. MR 2235828.
  35. R. Schaback, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math. 3 (1995), no. 3, 251–264. MR 1325034.
  36. G. B. Wright and B. Fornberg, Stable computations with flat radial basis functions using vector-valued rational approximations, J. Comput. Phys. 331 (2017), 137–156. MR 3588686.