FAQ

The adaptation of the cross validation aproach for RBF-based collocation methods

Publication date: 18.07.2017

Technical Transactions, 2017, Volume 7 Year 2017 (114), pp. 147 - 156

https://doi.org/10.4467/2353737XCT.17.115.6656

Authors

,
Artur Krowiak
Institute of Applied Informatics, Faculty of Mechanical Engineering, Cracow University of Technology
All publications →
Jordan Podgórski
Cracow University of Technology
All publications →

Titles

The adaptation of the cross validation aproach for RBF-based collocation methods

Abstract

W artykule pokazano adaptację algorytmu krzyżowego sprawdzania, znanego z zagadnień statystyki i interpolacji, do wyznaczenia wartości współczynnika kształtu w radialnych funkcjach bazowych. Funkcje te użyto w dwóch typach technik kolokacyjnych stosowanych na nieregularnej siatce do przybliżonego rozwiązywania równań różniczkowych. Aby otrzymać rezultaty o odpowiedniej dokładności, współczynnik kształtu powinien być dobrany na bazie kompromisu pomiędzy dokładnością a uwarunkowaniem układu równań. Przedstawiony algorytm, zwany “leave one out”, bierze te aspekty pod uwagę. Podsumowaniem artykułu są numeryczne testy, które pokazują użyteczność tego podejścia.

References

[1] Belytschko T., Krongauz Y., Organ D., Flrming M., Krysl P., Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering, vol. 139, 1996, 3-47.
[2] Liu G.R., Meshlees Methods – Moving beyond the Finite Element Method, CRC Press, Boca Raton, Florida 2003.
[3] Kansa E., Multiquadrics – A scattered data approximation scheme with applications to computational fluid dynamics I: Surface approximations and partial derivative estimates, Computers and Mathematics with Applications, vol. 19, 1990, 127-145.
[4] Kansa E., Multiquadrics – A scattered data approximation scheme with applications to computational fluid dynamics I: Solutions to parabolic, hyperbolic, and elliptic partial differential equations, Computers and Mathematics with Applications, vol. 19, 1990, 147-161.
[5] Fasshauer G.E., Meshfree Approximation Methods with Matlab, World Scientific Publishing, Singapore, 2007.
[6] Cheng A.H.D., Multiquadrics and its shape parameter – a numerical investigation of error estimate, condition number and round-off error by arbitrary precision computation, Engineering analysis with boundary elements, vol. 36, 2012, 220-239.
[7] Ferreira A.J.M, A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates, Composit Structures, vol. 59, 2003, 385-92.
[8] Schaback R., Error estimates and condition numbers for radial basis function interpolation, Advances in Computational Mathematics, vol. 3, 1995, 251-264.
[9] Krowiak A., Radial basis function-based pseudospectral method for static analysis of thin plates, Engineering Analysis with Boundary Elements, vol. 71, 2016, 50-58.
[10] Krowiak A., On choosing a value of shape parameter in Radial Basis Function collocation methods, Numerical Methods for Partial Differential Equations, submitted for publication.
[11] Hon Y.C., Schaback R., On nonsymmetric collocation by radial basis functions, Appl. Math. Comput., vol. 119, 2001, 177-186.
[12] Chen W., Fu Z.J., Chen C.S., Recent Advances in Radial Basis Function Collocation Methods, Springer, 2014.
[13] Rippa S., An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. in Comput. Math., vol. 11, 1999, 193-210.

Information

Information: Technical Transactions, 2017, Volume 7 Year 2017 (114), pp. 147 - 156

Article type: Original article

Titles:

Polish:

The adaptation of the cross validation aproach for RBF-based collocation methods

English:

The adaptation of the cross validation aproach for RBF-based collocation methods

Authors

Institute of Applied Informatics, Faculty of Mechanical Engineering, Cracow University of Technology

Cracow University of Technology

Published at: 18.07.2017

Article status: Open

Licence: None

Percentage share of authors:

Artur Krowiak (Author) - 50%
Jordan Podgórski (Author) - 50%

Article corrections:

-

Publication languages:

English