Next Article in Journal
Sliding Mode Control Algorithms for Anti-Lock Braking Systems with Performance Comparisons
Previous Article in Journal
Improved Sliding Mode Finite-Time Synchronization of Chaotic Systems with Unknown Parameters
Open AccessArticle

New Approach for Radial Basis Function Based on Partition of Unity of Taylor Series Expansion with Respect to Shape Parameter

1
Department of Mechanical Engineering, College of Engineering and Islamic Architecture, Umm Al-Qura University, P.O.Box 5555, Makkah 24382, Saudi Arabia
2
Department of Mechanical and Manufacturing Engineering, University of Calgary, 2500 University Drive, NW, Calgary, AB T2N 1N4, Canada
*
Author to whom correspondence should be addressed.
Received: 15 November 2020 / Revised: 15 December 2020 / Accepted: 16 December 2020 / Published: 22 December 2020
Radial basis function (RBF) is gaining popularity in function interpolation as well as in solving partial differential equations thanks to its accuracy and simplicity. Besides, RBF methods have almost a spectral accuracy. Furthermore, the implementation of RBF-based methods is easy and does not depend on the location of the points and dimensionality of the problems. However, the stability and accuracy of RBF methods depend significantly on the shape parameter, which is primarily impacted by the basis function and the node distribution. At a small value of shape parameter, the RBF becomes more accurate, but unstable. Several approaches were followed in the open literature to overcome the instability issue. One of the approaches is optimizing the solver in order to improve the stability of ill-conditioned matrices. Another approach is based on searching for the optimal value of the shape parameter. Alternatively, modified bases are used to overcome instability. In the open literature, radial basis function using QR factorization (RBF-QR), stabilized expansion of Gaussian radial basis function (RBF-GA), rational radial basis function (RBF-RA), and Hermite-based RBFs are among the approaches used to change the basis. In this paper, the Taylor series is used to expand the RBF with respect to the shape parameter. Our analyses showed that the Taylor series alone is not sufficient to resolve the stability issue, especially away from the reference point of the expansion. Consequently, a new approach is proposed based on the partition of unity (PU) of RBF with respect to the shape parameter. The proposed approach is benchmarked. The method ensures that RBF has a weak dependency on the shape parameter, thereby providing a consistent accuracy for interpolation and derivative approximation. Several benchmarks are performed to assess the accuracy of the proposed approach. The novelty of the present approach is in providing a means to achieve a reasonable accuracy for RBF interpolation without the need to pinpoint a specific value for the shape parameter, which is the case for the original RBF interpolation. View Full-Text
Keywords: radial basis function (RBF); taylor series; partition of unity; PUM; interpolation radial basis function (RBF); taylor series; partition of unity; PUM; interpolation
Show Figures

Figure 1

MDPI and ACS Style

Bawazeer, S.A.; Baakeem, S.S.; Mohamad, A.A. New Approach for Radial Basis Function Based on Partition of Unity of Taylor Series Expansion with Respect to Shape Parameter. Algorithms 2021, 14, 1. https://0-doi-org.brum.beds.ac.uk/10.3390/a14010001

AMA Style

Bawazeer SA, Baakeem SS, Mohamad AA. New Approach for Radial Basis Function Based on Partition of Unity of Taylor Series Expansion with Respect to Shape Parameter. Algorithms. 2021; 14(1):1. https://0-doi-org.brum.beds.ac.uk/10.3390/a14010001

Chicago/Turabian Style

Bawazeer, Saleh A.; Baakeem, Saleh S.; Mohamad, Abdulmajeed A. 2021. "New Approach for Radial Basis Function Based on Partition of Unity of Taylor Series Expansion with Respect to Shape Parameter" Algorithms 14, no. 1: 1. https://0-doi-org.brum.beds.ac.uk/10.3390/a14010001

Find Other Styles
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Search more from Scilit
 
Search
Back to TopTop