Abstract
Radial basis functions (RBFs) are isotropic, simple in form, dimensionally independent and mesh-free and are suitable for interpolation and fitting of scattered data. In a scattered point set, the calculation accuracy of multiquadric (MQ) RBF interpolation is strongly related to the selection of the shape factor. There is still no uniform method for determining the shape factor. Many scholars focus on determining the single optimal shape factor and seldom consider the change in the shape factor with the spatial point density in scattered point sets. In this paper, an adaptive radial basis function (ARBF) interpolation algorithm is proposed. The shape factors of MQ functions are determined adaptively by the local point densities of the points to be interpolated. To evaluate the computational performance of the ARBF interpolation algorithm, twelve groups of benchmark tests are conducted in this paper. We found that (1) the numerical error of ARBF interpolation is approximately 10% less than that of commonly used RBF interpolation with the shape factor recommended by Hardy. (2) The computational efficiency of ARBF interpolation is 1–2.5% lower than that of commonly used RBF interpolation with the shape factor recommended by Hardy.
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Bagheri S, Konen W, Emmerich M, Back T (2017) Self-adjusting parameter control for surrogate-assisted constrained optimization under limited budgets. Appl Soft Comput 61:377–393. https://doi.org/10.1016/j.asoc.2017.07.060
Barone GB, Boccia V, Bottalico D, Campagna R, Carracciuolo L, Laccetti G, Lapegna M (2017) An approach to forecast queue time in adaptive scheduling: how to mediate system efficiency and users satisfaction. Int J Parallel Prog 45(5):1164–1193. https://doi.org/10.1007/s10766-016-0457-y
Bayona V, Moscoso M, Kindelan M (2011) Optimal constant shape parameter for multiquadric based RBF-FD method. J Comput Phys 230(19):7384–7399. https://doi.org/10.1016/j.jcp.2011.06.005
Boyd JP (2010) Error saturation in Gaussian radial basis functions on a finite interval. J Comput Appl Math 234(5):1435–1441. https://doi.org/10.1016/j.cam.2010.02.019
Cheng AHD, Golberg MA, Kansa EJ, Zammito G (2003) Exponential convergence and h-c multiquadric collocation method for partial differential equations. Numer Methods Partial Differ Equ 19(5):571–594. https://doi.org/10.1002/num.10062
Cuomo S, Galletti A, Giunta G, Starace A (2013) Surface reconstruction from scattered point via RBF interpolation on GPU. In: Federated conference on computer science and information systems, pp 433–440
Cuomo S, Galletti A, Giunta G, Marcellino L (2017) Reconstruction of implicit curves and surfaces via RBF interpolation. Appl Numer Math 116:157–171. https://doi.org/10.1016/j.apnum.2016.10.016
Ding ZY, Mei G, Cuomo S, Xu NX, Tian H (2018) Performance evaluation of GPU-accelerated spatial interpolation using radial basis functions for building explicit surfaces. Int J Parallel Prog 46(5):963–991. https://doi.org/10.1007/s10766-017-0533-6
Ding Z, Mei G, Cuomo S, Li Y, Xu N (2020) Comparison of estimating missing values in iot time series data using different interpolation algorithms. Int J Parallel Prog 48(3):534–548. https://doi.org/10.1007/s10766-018-0595-5
Duchon J (1977) Splines minimizing rotation invariant seminorms in Sobolev spaces. In: Constructive theory of functions of several variables
Fasshauer GE, Zhang JG (2007) On choosing “optimal” shape parameters for RBF approximation. Numer Algorithms 45(1–4):345–368. https://doi.org/10.1007/s11075-007-9072-8
Feng R, Peng S (2018) Quasi-interpolation scheme for arbitrary dimensional scattered data approximation based on natural neighbors and rbf interpolation. J Comput Appl Math 329:95–105. https://doi.org/10.1016/j.cam.2017.02.026
Fornberg B, Wright G (2004) Stable computation of multiquadric interpolants for all values of the shape parameter. Comput Math Appl 48(5–6):853–867. https://doi.org/10.1016/j.camwa.2003.08.010
Franke R (1982) Scattered data interpolation—tests of some methods. Math Comput 38(157):181–200. https://doi.org/10.2307/2007474
Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76(8):1905–1915
Huang DS (1997) The united adaptive learning algorithm for the link weights and shape parameter in RBFN for pattern recognition. Int J Pattern Recognit Artif Intell 11(6):873–888. https://doi.org/10.1142/s0218001497000391
Huang CS, Lee CF, Cheng AHD (2007) Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method. Eng Anal Bound Elem 31(7):614–623. https://doi.org/10.1016/j.enganabound.2006.11.011
Iske A (2002) Scattered data modelling using radial basis functions. In: Tutorials on multiresolution in geometric modelling
Kazemi BF, Ghoreishi F (2013) Error estimate in fractional differential equations using multiquadratic radial basis functions. J Comput Appl Math 245:133–147. https://doi.org/10.1016/j.cam.2012.12.011
Krige DG (1951) A statistical approach to some mine valuations and allied problems at the Witwatersrand
Larsson E, Fornberg B (2005) Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Comput Math Appl 49(1):103–130. https://doi.org/10.1016/j.camwa.2005.01.010
Lu GY, Wong DW (2008) An adaptive inverse-distance weighting spatial interpolation technique. Comput Geosci 34(9):1044–1055. https://doi.org/10.1016/j.cageo.2007.07.010
Mei G, Xu N, Xu L (2016) Improving GPU-accelerated adaptive IDW interpolation algorithm using fast kNN search. SpringerPlus 5(1):1–22. https://doi.org/10.1186/s40064-016-3035-2
Nunez L, Regis RG, Varela K (2018) Accelerated random search for constrained global optimization assisted by radial basis function surrogates. J Comput Appl Math 340:276–295. https://doi.org/10.1016/j.cam.2018.02.017
Rippa S (1999) An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv Comput Math 11(2–3):193–210. https://doi.org/10.1023/a:1018975909870
Romani L, Rossini M, Schenone D (2019) Edge detection methods based on RBF interpolation. J Comput Appl Math 349:532–547. https://doi.org/10.1016/j.cam.2018.08.006
Wang JG, Liu GR (2002) On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Comput Methods Appl Mech Eng 191(23–24):2611–2630. https://doi.org/10.1016/s0045-7825(01)00419-4
Acknowledgements
This research was jointly supported by the National Natural Science Foundation of China (Grant Numbers: 11602235 and 41772326), and the Fundamental Research Funds for China Central Universities (Grant Numbers: 2652018091, 2652018107 and 2652018109). The authors would like to thank the editor and the reviewers for their contribution.
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Communicated by Yaroslav D. Sergeyev.
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Gao, K., Mei, G., Cuomo, S. et al. ARBF: adaptive radial basis function interpolation algorithm for irregularly scattered point sets. Soft Comput 24, 17693–17704 (2020). https://doi.org/10.1007/s00500-020-05211-0
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DOI: https://doi.org/10.1007/s00500-020-05211-0