Stable interpolation with isotropic and anisotropic Gaussians using Hermite generating function. (English) Zbl 07141501
Summary: Gaussian kernels can be an efficient and accurate tool for multivariate interpolation. For smooth functions, high accuracies are often achieved near the flat limit where the interpolation matrix becomes increasingly ill-conditioned. Stable evaluation algorithms for isotropic Gaussians (Gaussian radial basis functions) have been proposed based on a Chebyshev expansion of the Gaussians by B. Fornberg et al. [ibid. 33, No. 2, 869–892 (2011; Zbl 1227.65018)] and based on a Mercer expansion with Hermite polynomials by Fasshauer and McCourt. In this paper, we propose a new stabilization algorithm for the multivariate interpolation with isotropic or anisotropic Gaussians for an arbitrary number of dimensions derived from the generating function of the Hermite polynomials. We also derive and analyze a new analytic cutoff criterion for the generating function expansion that allows us to automatically adjust the number of stabilizing basis functions.
MSC:
| 65D05 | Numerical interpolation |
| 65D15 | Algorithms for approximation of functions |
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
| 41A63 | Multidimensional problems |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
radial basis functions; anisotropic Gaussians; stable evaluation; Hermite polynomials; generating functionsCitations:
Zbl 1227.65018References:
| [1] | M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Vol. 55, U.S. Government Printing Office, Washington, DC, 1964. · Zbl 0171.38503 |
| [2] | R. Beatson, O. Davydov, and J. Levesley, Error bounds for anisotropic RBF interpolation, J. Approx. Theory, 162 (2010), pp. 512-527. · Zbl 1194.41004 |
| [3] | G. Cheng and V. Shcherbakov, Anisotropic radial basis function methods for continental size ice sheet simulations, J. Comput. Phys., 372 (2018), pp. 161-177. · Zbl 1415.76497 |
| [4] | H. Dietert, J. Keller, and S. Troppmann, An invariant class of wave packets for the Wigner transform, J. Math. Anal. Appl., 450 (2017), pp. 1317-1332. · Zbl 1376.81018 |
| [5] | G. E. Fasshauer, F. J. Hickernell, and H. Woźniakowski, On dimension-independent rates of convergence for function approximation with Gaussian kernels, SIAM J. Numer. Anal., 50 (2012), pp. 247-271. · Zbl 1243.65025 |
| [6] | G. E. Fasshauer and M. McCourt, Kernel-Based Approximation Methods using Matlab, Vol. 19, World Scientific, River Edge, NJ, 2015. · Zbl 1318.00001 |
| [7] | G. E. Fasshauer and M. J. McCourt, Stable evaluation of Gaussian radial basis function interpolants, SIAM J. Sci. Comput., 34 (2012), pp. A737-A762. · Zbl 1252.65028 |
| [8] | B. Fornberg and N. Flyer, Solving PDEs with radial basis functions, Acta Numer., 24 (2015), p. 215-258. · Zbl 1316.65073 |
| [9] | B. Fornberg, E. Larsson, and N. Flyer, Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput., 33 (2011), pp. 869-892. · Zbl 1227.65018 |
| [10] | B. Fornberg, E. Lehto, and C. Powell, Stable calculation of Gaussian-based RBF-FD stencils, Comput. Math. Appl., 65 (2013), pp. 627-637. · Zbl 1319.65011 |
| [11] | B. Fornberg and C. Piret, A stable algorithm for flat radial basis functions on a sphere, SIAM J. Sci. Comput., 30 (2007), pp. 60-80. · Zbl 1159.65307 |
| [12] | B. Fornberg and G. B. Wright, Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl., 48 (2004), pp. 853-867. · Zbl 1072.41001 |
| [13] | B. Fornberg and J. Zuev, The Runge phenomenon and spatially variable shape parameters in RBF interpolation, Comput. Math. Appl., 54 (2007), pp. 379-398. · Zbl 1128.41001 |
| [14] | M. Griebel, C. Rieger, and B. Zwicknagl, Multiscale approximation and reproducing kernel Hilbert space methods, SIAM J. Numer. Anal., 53 (2015), pp. 852-873. · Zbl 1312.41025 |
| [15] | G. A. Hagedorn, Generating function and a Rodrigues formula for the polynomials in d-dimensional semiclassical wave packets, Ann. Physics, 362 (2015), pp. 603-608. · Zbl 1343.81100 |
| [16] | E. Larsson and B. Fornberg, Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions, Comput. Math. Appl., 49 (2005), pp. 103-130. · Zbl 1074.41012 |
| [17] | E. Larsson, E. Lehto, A. Heryudono, and B. Fornberg, Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions, SIAM J. Sci. Comput., 35 (2013), pp. A2096-A2119. · Zbl 1362.65026 |
| [18] | E. Larsson, V. Shcherbakov, and A. Heryudono, A least squares radial basis function partition of unity method for solving PDEs, SIAM J. Sci. Comput., 39 (2017), pp. A2538-A2563. · Zbl 1377.65156 |
| [19] | S. De Marchi and G. Santin, A new stable basis for radial basis function interpolation, J. Comput. Appl. Math., 253 (2013), pp. 1-13. · Zbl 1288.65013 |
| [20] | M. McCourt and G. E. Fasshauer, Stable likelihood computation for Gaussian random fields, in Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science, Springer, Cham, Switzerland, 2017, pp. 917-943. · Zbl 1408.62158 |
| [21] | J. Rashidinia, G. E. Fasshauer, and M. Khasi, A stable method for the evaluation of Gaussian radial basis function solutions of interpolation and collocation problems, Comput. Math. Appl., 72 (2016), pp. 178-193. · Zbl 1443.65110 |
| [22] | A. E. Tarwater, Parameter Study of Hardy’s Multiquadric Method for Scattered Data Interpolation, Technical report UCRL-53670, Lawrence Livermore National Laboratory, Livermore, CA, 1985. |
| [23] | G. Watson, Notes on generating functions of polynomials: (2) Hermite polynomials, J. Lond. Math. Soc. (2), 1 (1933), pp. 194-199. · Zbl 0007.20301 |
| [24] | G. B. Wright and B. Fornberg, Stable computations with flat radial basis functions using vector-valued rational approximations, J. Comput. Phys., 331 (2017), pp. 137-156. · Zbl 1378.65045 |
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