Current issue
Archive
About the Journal
About
Editorial Board
Abstract & Indexing
Contact
Editorial Policies
Authorship & COI
Ethical principles
Principles of Transparent Checklist
Open access
For Authors
Instructions to Authors
Detailing Guidelines for Authors
Review process
Review sheet
How to submit
Open Access license
ORIGINAL PAPER
Residual error based adaptive method with an optimal variable scaling parameter for RBF interpolation
1
Department of Mathematics, Osmania University College for Women, Hyderabad, INDIA
2
Department of Mathematics, Ecole Centrale School of Engineering
Mahindra University, Hyderabad, INDIA
3
Department of Mathematics, Osmania University, Hyderabad, INDIA
Publication date: 2023-03-01
International Journal of Applied Mechanics and Engineering 2023;28(1):37-46
KEYWORDS
ABSTRACT
In infinitely smooth Radial Basis Function (RBF) based interpolation, the scaling parameter plays an important role to obtain an accurate and stable numerical solution. When this method is applied to interpolate a function with sharp gradients, then adaptive methods will also play a significant role in determining an optimal number of centers according to the user desired accuracy. In this article, we test an optimization algorithm developed using the nonlinear optimization to find a scaling parameter for RBF along with an adaptive residual subsampling method [1] RBF interpolation. In this process, at each stage of adoption, the available optimal shape parameters have been obtained by solving the system of non-linear equations.
REFERENCES (39)
1.
Driscoll Tobin A. and Heryudono Alfa R.H.(2007): Adaptive residual subsampling method for radial basis function interpolation and collocation problems.– Comput. Math. Appl., vol.53, pp.927-939.
2.
Buhmann M.D. (2003): Radial Basis Functions.– Cambridge University Press, Cambridge.
3.
Dyn N. (1987): Interpolation of scattered data by radial basis function.– In: Topics in Multivariate Approximation, Eds. C.K. Chui, L.L. Schumarket and F.I. Utreras, Academic Press, New York, pp.47-61.
4.
Franke R. (1982): Scattered data interpolation: tests of some methods.– Math. Comput., vol.38, pp.181-200.
5.
Micchelli C.A. (1986): Interpolation of scattered data: distance matrices and conditionally positive definite functions.– Constructive Approximation, vol.2, pp.11-12.
6.
Cheng A.H.D. (2012): Multiquadric and its shape parameter. A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation.– Eng. Anal. Bound. Elements, vol.36, pp.220-239.
7.
Fasshauer G.E. and Zhang, J.G. (2007): On choosing “optimal” shape parameters for RBF approximation.– Numer. Algorithms, vol.45, pp.345-368.
8.
Fornberg B. and Wright G. (2004): Stable computation of multiquadric interpolants for all values of the shape parameter.– Comput. Math. Appl., vol.48, No.5-6, pp.853-867.
9.
Rippa S. (1999): An algorithm for selecting a good value for the parameter c in radial basis function interpolation.– Adv. Comput. Math., vol.11, pp.193-210.
10.
Kansa E.J. (1990): Multiquadrics a scattered data approximation scheme with applications to computational fluid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations.– Comput. Math. Appl., vol.19, pp.147-161.
11.
Chen W., Fu Z.J. and Chen C.S. (2014): Recent Advances in Radial Basis Function Collocation Methods.– Springer Briefs in Applied Sciences and Technology, Springer, Heidelberg.
12.
Kansa E.J. and Carlson R.E. (1992): Improved accuracy of multiquadric interpolation using variable shape parameters.– Comput. Math. Appl., vol.24, pp.99-120.
13.
Carlson R.E. and Foley T.A. (1991): The parameter R2 in multiquadric interpolation.– Comput. Math. Appl., vol.21, pp.29-42.
14.
Hardy R.L. (1971): Multiquadric equations of topography and other irregular surfaces.– J. Geophys. Res., vol.76, pp.1905-1915.
15.
Foley T.A. (1994): Near Optimal Parameter Selection for Multiquadric Interpolation.– Manuscript, Computer Science and Engineering Department, Arizona State University, Tempe.
16.
Scheuerer M. (2011): An alternative procedure for selecting a good value for the parameter c in RBF-interpolation.– Adv. Comput. Math., vol.34, pp.105-126.
17.
Roque C. and Ferreira A.J. (2010): Numerical experiments on optimal shape parameters for radial basis functions.– Numer. Meth. Partial Differ. Eq., vol.26, pp.675-689.
18.
Sanyasiraju Y.V.S.S. and Satyanarayana C. (2013): On optimization of the RBF shape parameter in a grid-free local scheme for convection dominated problems over non-uniform centers.– Appl. Math. Model., vol.37, pp.7245-7272.
19.
Larsson E. and Fornberg B. (2005): Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions.– Comput. Math. Appl., vol.49, pp.103-130.
20.
Fornberg B., Larsson E. and Flyer N. (2011): Stable computations with Gaussian radial basis functions.– SIAM J. Sci. Comput., vol.33, pp.869-892.
21.
Luh L.T. (2014): The mystery of the shape parameter IV.– Eng. Anal. Bound. Elem., vol.48, pp.24-31.
22.
Cavoretto R., De Rossi A. and Perracchione E. (2017): Optimal selection of local approximants in RBF-PU interpolation.– J. Sci. Comput., vol.74, No.1, pp.1-22.
23.
Jafar Biazar and Mohammad Hosami (2017): An interval for the shape parameter in radial basis function approximation.– Comput. Math. Appl., vol.315, pp.131-149.
24.
Jankowska M.A., Karageorghis A. and Chen C.S. (2018): Kansa RBF method for nonlinear problem.– Int. J. Comp. Meth. and Exp. Meas., vol.6, No.6, pp.1000-1007.
25.
Jankowska M.A. and Karageorghis A. (2019): Variable shape parameter Kansa RBF method for the solution of nonlinear boundary value problems.– Eng. Anal. Bound. Elements., vol.103, pp.32-40.
26.
Hon Y.C. (1999): Multiquadric collocation method with adaptive technique for problems with boundary layer.– Internat. J. Appl. Sci. Comput., vol.6, No.3, pp.173-184.
27.
Schaback R. and Wendland H. (2000): Adaptive greedy techniques for approximate solution of large RBF systems.– Numer. Algorithms, vol.24, No.3, pp.239-254.
28.
Hon Y.C., Schaback R. and Zhou X. (2003): An adaptive greedy algorithm for solving large RBF collocation problems.– Numer. Algorithms. vol.32, No.1, pp.13-25.
29.
Bozzini M., Lenarduzzi L. and Schaback R. (2002): Adaptive interpolation by scaled multiquadrics.– Adv. Comput. Math., vol.16, No.4, pp.375-387.
30.
Behrens J., Iske A. and Kaser, M. (2003): Adaptive meshfree method of backward characteristics for nonlinear transport equations.– In: Meshfree Methods for Partial Differential Equations (Bonn, 2001), in: Lect. Notes Comput. Sci. Eng., vol.26, Springer, pp.21-36.
31.
Behrens, J., Iske, A. (2002): Grid-free adaptive semi- Lagrangian advection using radial basis functions.– Comput. Math. Appl. vol.43, No.3-5, pp.319-327.
32.
Sarra S.A. (2005): Adaptive radial basis function methods for time dependent partial differential equations.– Appl. Numer. Math., vol.54, No.1, pp.79-94.
33.
Cavoretto R. and De Rossi A. (2019): Adaptive meshless refinement scheme for RBF- PUM collocation.– Appl. Math. Letter., vol.90, pp.131-138.
34.
Cavoretto R. and De Rossi A. (2020): An adaptive LOOCV based algorithm for solving elliptic PDEs via RBF collocation.– In book: Large scale scientific computing, pp.76-86.
35.
Cavoretto R. and De Rossi A. (2020): A two stage adaptive scheme based on RBF collocation for solving elliptical PDEs.– Comput. Math. Appl., vol.79, pp.3206-3222.
36.
Cavoretto R. and De Rossi A. (2020): Adaptive procedure for meshfree RBF unsymmetric and symmetric collocation methods.– Applied Math. and Computation, Elsevier, vol.382.
37.
Fasshauer G.E. (2007): Meshfree Approximation Methods with MATLAB.– World Scientific.
38.
Wendland H. (2005): Proceedings of the Scattered Data approximation.– Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, vol.17, pp.28-29.
39.
Fornberg B. and Wright G. (2004): Stable computation of multiquadric interpolants for all values of the shape parameter.– Comput. Math. Appl., vol.48, pp.853-867.
| eISSN: | 2353-9003 |
| ISSN: | 1734-4492 |
Task: edition of a scientific journal, its internationalization and ensuring open access to the journal via the Internet
© 2006-2024 Journal hosting platform by Bentus
We process personal data collected when visiting the website. The function of obtaining information about users and their behavior is carried out by voluntarily entered information in forms and saving cookies in end devices. Data, including cookies, are used to provide services, improve the user experience and to analyze the traffic in accordance with the Privacy policy. Data are also collected and processed by Google Analytics tool (more).
You can change cookies settings in your browser. Restricted use of cookies in the browser configuration may affect some functionalities of the website.
You can change cookies settings in your browser. Restricted use of cookies in the browser configuration may affect some functionalities of the website.
