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Kernel-based adaptive approximation of functions with discontinuities. (English) Zbl 1411.65024

Summary: One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in those subdomains, and these sub-approximations can possibly be calculated efficiently in parallel, as long as the subdomains do not overlap. This paper proposes an algorithm that first calculates sub-approximations on non-overlapping subdomains, then extends the subdomains as much as possible and finally produces a global solution on the given domain by letting the subdomains fill the whole domain. Consequently, there will be no Gibbs phenomenon along the boundaries of the subdomains. The method detects faults and gradient faults with good accuracy. Throughout, the algorithm works for fixed scattered input data of the function itself, not on spectral data, and it does not resample.

MSC:

65D05 Numerical interpolation
41A05 Interpolation in approximation theory
62H30 Classification and discrimination; cluster analysis (statistical aspects)
68T05 Learning and adaptive systems in artificial intelligence

Software:

GaussQR
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Full Text: DOI arXiv

References:

[1] Allasia, G.; Besenghi, R.; Cavoretto, R., Adaptive detection and approximation of unknown surface discontinuities from scattered data, Simul. Model. Pract. Theory, 17, 1059-1070 (2009)
[2] Arandiga, F.; Cohen, A.; Donat, R.; Dyn, N., Interpolation and approximation of piecewise smooth functions, SIAM J. Numer. Anal., 43, 41-57 (2005) · Zbl 1092.65004
[3] Bentley, J. L., Multidimensional binary search trees used for associative searching, Commun. ACM, 18, 509-517 (1975) · Zbl 0306.68061
[4] Bozzini, M.; Lenarduzzi, L.; Rossini, M., Non-regular surface approximation, (Floater, M., Mathematical Methods for Curves and Surfaces. Mathematical Methods for Curves and Surfaces, Lecture Notes in Computer Science, 8177 (2012)), 68-87 · Zbl 1356.65044
[5] Bozzini, M.; Rossini, M., The detection and recovery of discontinuity curves from scattered data, J. Comput. Appl. Math., 240, 148-162 (2013) · Zbl 1255.65047
[6] Buhmann, M. D., Radial Basis Functions, Theory and Implementations (2003), Cambridge University Press · Zbl 1038.41001
[7] Cristianini, N.; Shawe-Taylor, J., An Introduction to Support Vector Machines and other Kernel-Based Learning Methods (2000), Cambridge University Press: Cambridge University Press Cambridge
[8] Fasshauer, G.; McCourt, M., Kernel-based Approximation Methods using MATLAB, Interdisciplinary Mathematical Sciences, 19 (2015), World Scientific: World Scientific Singapore
[9] Gutzmer, T.; Iske, A., Detection of discontinuities in scattered data approximation, Numer. Algorithm, 16, 155-170 (1997) · Zbl 0897.65004
[10] Harten, A., Multiresolution representation of data: a general framework, SIAM J. Numer. Anal., 33, 3, 1205-1256 (1996) · Zbl 0861.65130
[11] Kruskal, J. B., On the shortest spanning subtree of a graph and the traveling salesman problem, Proc. Amer. Math. Soc., 7, 48-50 (1956) · Zbl 0070.18404
[12] Lipman, Y.; Cohen-Or, D.; Levin, D., Data-dependent MLS for faithful surface approximation, (Belyaev, A.; Garland, M., Proceedings of the Fifth Eurographics Symposium on Geometry Processing (2007)), 59-67
[13] Lipman, Y.; Levin, D., Approximating piecewise smooth functions, IMA J. Numer. Anal., 30, 1159-1183 (2010) · Zbl 1222.65018
[14] de Silanes, M. C.L.; Parra, M. C.; Torrens, J. J., Vertical and oblique fault detection in explicit surfaces, J. Comput. Appl. Math., 140, 559-585 (2002) · Zbl 1019.65018
[15] de Silanes, M. C.L.; Parra, M. C.; Torrens, J. J., On a new characterization of finite jump discontinuities and its application to vertical fault detection, Math. Comput. Simul., 77, 247-256 (2008) · Zbl 1143.65012
[16] Narcowich, F. J.; Ward, J. D.; Wendland, H., Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comput., 74, 743-763 (2005) · Zbl 1063.41013
[18] Schaback, R.; Wendland, H., Kernel techniques: from machine learning to meshless methods, Acta Numerica, 15, 543-639 (2006) · Zbl 1111.65108
[19] Schaback, R.; Werner, J., Linearly constrained reconstruction of functions by kernels with applications to machine learning, Adv. Comput. Math., 25, 1-3, 237-258 (2006) · Zbl 1106.65008
[20] Schölkopf, B.; Smola, A. J., Learning with Kernels (2002), MIT Press: MIT Press Cambridge
[21] Shawe-Taylor, J.; Cristianini, N., Kernel Methods for Pattern Analysis (2004), Cambridge University Press
[22] Wendland, H., Scattered Data Approximation (2005), Cambridge University Press · Zbl 1075.65021
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