Article
| Title: | Geodesic metrics for RBF approximation of some physical quantities measured on sphere (English) |
| Author: | Segeth, Karel |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 69 |
| Issue: | 5 |
| Year: | 2024 |
| Pages: | 621-632 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points on a sphere. The choice of a particular geodesic metric function is an important part of the construction of interpolation formula. We show the existence of an interpolation formula of the type considered. The approximation formulas of this type can be useful in the interpretation of measurements of various physical quantities. We present an example concerned with the sampling of anisotropy of magnetic susceptibility having extensive applications in geosciences and demonstrate the advantages and drawbacks of the formulas chosen, in particular the strong dependence of interpolation results on condition number of the matrix of the system considered and on round-off errors in general. (English) |
| Keyword: | spherical interpolation |
| Keyword: | radial basis function |
| Keyword: | spherical radial basis function |
| Keyword: | geodesic metric |
| Keyword: | trend |
| Keyword: | multiquadric |
| Keyword: | magnetic susceptibility |
| MSC: | 65D05 |
| MSC: | 65D10 |
| MSC: | 65D12 |
| MSC: | 65Z05 |
| DOI: | 10.21136/AM.2024.0051-24 |
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| Date available: | 2024-11-05T12:01:47Z |
| Last updated: | 2024-11-05 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152634 |
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| Reference: | [1] Baxter, B. J., Hubbert, S.: Radial basis functions for the sphere.Recent Progress in Multivariate Approximation International Series of Numerical Mathematics 137. Birkhäuser, Basel (2001), 33-47. MR 1877496, 10.1007/978-3-0348-8272-9_4 |
| Reference: | [2] Buhmann, M. D.: Radial Basis Functions: Theory and Implementations.Cambridge Monographs on Applied and Computational Mathematics 12. Cambridge University Press, Cambridge (2003). Zbl 1038.41001, MR 1997878, 10.1017/CBO9780511543241 |
| Reference: | [3] Duchon, J.: Sur l'erreur d'interpolation des fonctions de plusieurs variable par les $D^m$-splines.RAIRO, Anal. Numér. 12 (1978), 325-334 French. Zbl 0403.41003, MR 0519016, 10.1051/m2an/1978120403251 |
| Reference: | [4] Fiedler, M.: Special Matrices and Their Applications in Numerical Mathematics.Martinus Nijhoff Publishers, Dordrecht (1986). Zbl 0677.65019, MR 1105955 |
| Reference: | [5] Golub, G. H., Loan, C. F. Van: Matrix Computations.Johns Hopkins University Press, Baltimore (1996). Zbl 0865.65009, MR 1417720 |
| Reference: | [6] Hrouda, F., Ježek, J., Chadima, M.: On the origin of apparently negative minimum susceptibility of hematite single crystals calculated from low-field anisotropy of magnetic susceptibility.Geophys. J. Int. 224 (2021), 1905-1917. 10.1093/gji/ggaa546 |
| Reference: | [7] Hubbert, S., Gia, Q. T. Lê, Morton, T. M.: Spherical Radial Basis Functions: Theory and Applications.SpringerBriefs in Mathematics. Springer, Cham (2015). Zbl 1321.33017, MR 3330978, 10.1007/978-3-319-17939-1 |
| Reference: | [8] Levesley, J., Luo, Z., Sun, X.: Norm estimates of interpolation matrices and their inverses associated with strictly positive definite functions.Proc. Am. Math. Soc. 127 (1999), 2127-2134. Zbl 0924.65006, MR 1476145, 10.1090/S0002-9939-99-04683-3 |
| Reference: | [9] Micchelli, C. A.: Interpolation of scattered data: Distance matrices and conditionally positive definite functions.Constr. Approx. 2 (1986), 11-22. Zbl 0625.41005, MR 0891767, 10.1007/BF01893414 |
| Reference: | [10] Nagata, T.: Rock Magnetism.Maruzen, Tokyo (1961). |
| Reference: | [11] Segeth, K.: Some computational aspects of smooth approximation.Computing 95 (2013), S695--S708. MR 3054397, 10.1007/s00607-012-0252-6 |
| Reference: | [12] Segeth, K.: Spherical radial basis function approximation of some physical quantities measured.J. Comput. Appl. Math. 427 (2023), Article ID 115128, 7 pages. Zbl 1512.65032, MR 4549571, 10.1016/j.cam.2023.115128 |
| Reference: | [13] Tarling, D., Hrouda, F.: Magnetic Anisotropy of Rocks.Chapman and Hall, London (1993). |
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