# Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

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## Abstract

Scattered data interpolation schemes using kriging and radial basis functions (RBFs) have the advantage of being meshless and dimensional independent; however, for the datasets having insufficient observations, RBFs have the advantage over geostatistical methods as the latter requires variogram study and statistical expertise. Moreover, RBFs can be used for scattered data interpolation with very good convergence, which makes them desirable for shape function interpolation in meshless methods for numerical solution of partial differential equations. For interpolation of large datasets, however, RBFs in their usual form, lead to solving an ill-conditioned system of equations, for which, a small error in the data can cause a significantly large error in the interpolated solution. In order to reduce this limitation, we propose a hybrid kernel by using the conventional Gaussian and a shape parameter independent cubic kernel. Global particle swarm optimization method has been used to analyze the optimal values of the shape parameter as well as the weight coefficients controlling the Gaussian and the cubic part in the hybridization. Through a series of numerical tests, we demonstrate that such hybridization stabilizes the interpolation scheme by yielding a far superior implementation compared to those obtained by using only the Gaussian or cubic kernels. The proposed kernel maintains the accuracy and stability at small shape parameter as well as relatively large degrees of freedom, which exhibit its potential for scattered data interpolation and intrigues its application in global as well as local meshless methods for numerical solution of PDEs.

## Keywords

Radial basis function Multivariate interpolation Particle swarm optimization Spatial data analysis## Mathematics Subject Classification (2010)

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## References

- 1.Barnett, G.A.: A robust RBF-FD formulation based on polyharmonic splines and polynomials. Ph.D. thesis, University of Colarado, USA (2015)Google Scholar
- 2.Chen, W., Fu, Z., Chen, C.: Recent Advances in Radial Basis Function Collocation Methods. Springer, Berlin (2014)CrossRefGoogle Scholar
- 3.Driscoll, T.A., Fornberg, B.: Interpolation in the limit of increasingly flat radial basis functions. Comput. Math. Appl.
**43**, 413–422 (2002)CrossRefGoogle Scholar - 4.Eberhart, R., Kennedy, J.: A new optimizer using particle swarm theory. In: Proceedings of the Sixth International Symposium on Micro Machine and Human Science, 1995. MHS ’95, pp 39–43 (1995)Google Scholar
- 5.Eberhart, R., Shi, Y.: Particle swarm optimization: developments, applications and resources. In: Proceedings of the 2001 Congress on Evolutionary Computation, 2001, vol. 1, pp 81–86 (2001)Google Scholar
- 6.Fasshauer, G.F.: Meshfree Approximation Methods with MATLAB. World Scientific Publishing Co., Inc., River Edge (2007)CrossRefGoogle Scholar
- 7.Fasshauer, G.E., McCourt, M.J.: Stable evaluation of Gaussian radial basis function interpolants. SIAM J. Sci. Comput.
**34**(2), A737–A762 (2012)CrossRefGoogle Scholar - 8.Fasshauer, G.E., McCourt, M.: Kernel-Based Approximation Methods Using MATLAB. World Scientific, Interdisciplinary Mathematical Sciences (2015)Google Scholar
- 9.Fasshauer, G., Zhang, J.: Preconditioning of Radial Basis Function Interpolation Systems via Accelerated Iterated Approximate Moving Least Squares Approximation, Computational Methods in Applied Sciences, vol. 11. Springer, Netherlands (2009)Google Scholar
- 10.Flyer, N., Wright, G.B., Fornberg, B.: Handbook of Geomathematics, chap. Radial Basis Function-Generated Finite Differences: a Mesh-Free Method for Computational Geosciences, pp 1–30. Springer, Berlin (2014)Google Scholar
- 11.Fornberg, B., Flyer, N.: A Primer on Radial Basis Functions with Applications to the Geosciences. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2015)CrossRefGoogle Scholar
- 12.Fornberg, B., Piret, C.: A stable algorithm for flat radial basis functions on a sphere. SIAM J. Sci. Comput.
**30**(1), 60–80 (2007)CrossRefGoogle Scholar - 13.Fornberg, B., Driscoll, T., Wright, G., Charles, R.: Observations on the behavior of radial basis function approximations near boundaries. Computers & Mathematics with Applications
**43**(3–5), 473–490 (2002)CrossRefGoogle Scholar - 14.Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput.
**33**(2), 869–892 (2011)CrossRefGoogle Scholar - 15.Fornberg, B., Lehto, E., Powell, C.: Stable calculation of Gaussian-based RBF-FD stencils. Computers & Mathematics with Applications
**65**(4), 627–637 (2013)CrossRefGoogle Scholar - 16.Franke, R.: A Critical Comparison of Some Methods for Interpolation of Scattered Data. Final report. Defense Technical Information Center (1979)Google Scholar
- 17.Friedman, J., Hastie, T., Tibshirani, R.: The Elements of Statistical Learning, vol. 1. Springer series in statistics, New York (2001)Google Scholar
- 18.Getoor, L., Taskar, B.: Introduction to Statistical Relational Learning. MIT Press, Cambridge (2007)Google Scholar
- 19.Gonzalez-Rodriguez, P., Moscoso, M., Kindelan, M.: Laurent expansion of the inverse of perturbed, singular matrices. J. Comput. Phys.
**299**, 307–319 (2015)CrossRefGoogle Scholar - 20.Hardy, R.L.: Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res.
**76**(8), 1905–1915 (1971)CrossRefGoogle Scholar - 21.Kansa, E., Hon, Y.: Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Computers & Mathematics with Applications
**39**(7–8), 123–137 (2000)CrossRefGoogle Scholar - 22.Kindelan, M., Moscoso, M., González-Rodríguez, P.: Radial basis function interpolation in the limit of increasingly flat basis functions. J. Comput. Phys.
**307**, 225–242 (2016)CrossRefGoogle Scholar - 23.Lin, J., Chen, W., Sze, K.: A new radial basis function for Helmholtz problems. Engineering Analysis with Boundary Elements
**36**(12), 1923–1930 (2012)CrossRefGoogle Scholar - 24.Marchi, S.D., Santin, G.: A new stable basis for radial basis function interpolation. J. Comput. Appl. Math.
**253**, 1–13 (2013)CrossRefGoogle Scholar - 25.Mishra, P., Nath, S., Fasshauer, G., Sen, M., et al.: Frequency-domain meshless solver for acoustic wave equation using a stable radial basis-finite difference (RBF-FD) algorithm with hybrid kernels. In: 2017 SEG International Exposition and Annual Meeting. Society of Exploration Geophysicists (2017)Google Scholar
- 26.Mishra, P.K., Nath, S.K., Kosec, G., Sen, M.K.: An improved radial basis-pseudospectral method with hybrid gaussian-cubic kernels. Engineering Analysis with Boundary Elements
**80**, 162–171 (2017)CrossRefGoogle Scholar - 27.Perez, R., Behdinan, K.: Particle swarm approach for structural design optimization. Comput. Struct.
**85**(19–20), 1579–1588 (2007)CrossRefGoogle Scholar - 28.Rippa, S.: An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comput. Math.
**11**(2-3), 193–210 (1999)CrossRefGoogle Scholar - 29.Rusu, C., Rusu, V.: Artificial Intelligence in Theory and Practice: IFIP 19th World Computer Congress, TC 12: IFIP AI 2006 Stream, August 21–24, 2006, Santiago, Chile, chap. Radial Basis Functions Versus Geostatistics in Spatial Interpolations, pp 119–128. Springer, Boston (2006)Google Scholar
- 30.Sarra, S.A.: Radial basis function approximation methods with extended precision floating point arithmetic. Engineering Analysis with Boundary Elements
**35**(1), 68–76 (2011)CrossRefGoogle Scholar - 31.Sarra, S.A.: Regularized symmetric positive definite matrix factorizations for linear systems arising from RBF interpolation and differentiation. Engineering Analysis with Boundary Elements
**44**(7), 76–1245 (2014)CrossRefGoogle Scholar - 32.Sarra, S.A., Sturgill, D.: A random variable shape parameter strategy for radial basis function approximation methods. Engineering Analysis with Boundary Elements
**33**(11), 1239–1245 (2009)CrossRefGoogle Scholar - 33.Schaback, R.: Reproduction of Polynomials by Radial Basis Functions. Wavelets, Images, and Surface Fitting (1994)Google Scholar
- 34.Shaw, R., Srivastava, S.: Particle swarm optimization: a new tool to invert geophysical data. Geophysics
**72**(2), F75–F83 (2007)CrossRefGoogle Scholar - 35.Singh, A., Biswas, A.: Application of global particle swarm optimization for inversion of residual gravity anomalies over geological bodies with idealized geometries. Nat. Resour. Res.
**25**(3), 297–314 (2015)CrossRefGoogle Scholar - 36.Trauth, M.H.: MATLAB Recipes for Earth Sciences. Springer, Berlin (2010)CrossRefGoogle Scholar
- 37.Wahba, G.: Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 59. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1990)Google Scholar