Abstract
In some approximation problems, sampling from the target function can be both expensive and time-consuming. It would be convenient to have a method for indicating where approximation quality is poor, so that generation of new data provides the user with greater accuracy where needed. In this paper, we propose a new adaptive algorithm for radial basis function (RBF) interpolation which aims to assess the local approximation quality, and add or remove points as required to improve the error in the specified region. For Gaussian and multiquadric approximation, we have the flexibility of a shape parameter which we can use to keep the condition number of interpolation matrix at a moderate size. Numerical results for test functions which appear in the literature are given for dimensions 1 and 2, to show that our method performs well. We also give a three-dimensional example from the finance world, since we would like to advertise RBF techniques as useful tools for approximation in the high-dimensional settings one often meets in finance.
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Agnantiaris, J.P., Polyzos, D., Beskos, D. E: Some study on dual reciprocity BEM for elastodynamic analysis. Comput. Meach. 17, 270–277 (1996)
Atteia, M.: Fonctions spline et noyaux reproduisants d’Aronszajn-Bergman. Rev. Fran caise Informat. Recherche Op erationnelle 4, 31–43 (1970)
Ball, K., Sivakumar, N., Ward, J. D: On the sensitivity of radial basis interpolation to minimum point separation. J. App. Theory 8, 401–426 (1992)
Behrens, J., Iske, A.: Grid-free adaptive semi-Lagrangian advection using radial basis functions. Comput. Math. Appl. 43(3–5), 319—327 (2002)
Bozzini, M., Lenarduzzi, L., Schaback, R.: Adaptive interpolation by scaled multiquadrics. Adv. Comput. Math. 16, 375–387 (2002)
Bozzini, M., Lenarduzzi, L., Rossini, M., Schaback, R.: Interpolation with variably scaled kernels. IMA J. Numer. Anal. 35, 199–219 (2015)
Bozzini, M., Rossini, M.: Testing methods for 3D scattered data interpolation. Monografia de la Academia de Ciencias de Zaragoza 20, 111–135 (2002)
Driscoll, T. A., Heryudono, A. R. H.: Adaptive residual subsampling methods for radial basis function interpolation and collocation problems. Comput. Math. Appl. 53(6), 927–939 (2007)
Duchon, J: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976). Lecture Notes in Math., vol. 571, pp 85–100. Springer, Berlin (1977)
Hon, Y. C., Schaback, R., Zhou, X.: An adaptive greedy algorithm for solving large RBF collocation problems. Numer. Algor. 32(1), 13–25 (2003)
Gutzmer, T., Iske, A: Detection of discontinuities in scattered data approximation. Numer. Algor. 16, 155–170 (1997)
Iske, A., Levesley, J: Multilevel scattered data approximation by adaptive domain decomposition. Numer. Algor. 39, 187–198 (2005)
Levesley, J., Ragozin, D. L.: Local approximation on manifolds using radial functions and polynomials. In: International conference on curves and surfaces [4th], Saint-Malo, Proceedings, vol. 2. Curve and Surface Fitting, pp. 291–300 (1999)
Powell, M. J. D.: The theory of radial basis function approximation in 1990, advance in numerical analysis. In: Light, W. (ed.) , vol. II, pp 105–210. Oxford University Press (1992)
Schaback, R: Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 17, 270–277 (1995)
Schaback, R., Wendland, H: Adaptive greedy techniques for approximate solution of large RBF systems. Numer. Algor. 24(3), 239–254 (2000)
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Zhang, Q., Zhao, Y. & Levesley, J. Adaptive radial basis function interpolation using an error indicator. Numer Algor 76, 441–471 (2017). https://doi.org/10.1007/s11075-017-0265-5
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DOI: https://doi.org/10.1007/s11075-017-0265-5