{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T13:22:35Z","timestamp":1762608155258,"version":"build-2065373602"},"reference-count":55,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,22]],"date-time":"2020-12-22T00:00:00Z","timestamp":1608595200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/a14010001","type":"journal-article","created":{"date-parts":[[2020,12,22]],"date-time":"2020-12-22T12:42:28Z","timestamp":1608640948000},"page":"1","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["New Approach for Radial Basis Function Based on Partition of Unity of Taylor Series Expansion with Respect to Shape Parameter"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2095-3939","authenticated-orcid":false,"given":"Saleh A.","family":"Bawazeer","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, College of Engineering and Islamic Architecture, Umm Al-Qura University, P.O.Box 5555, Makkah 24382, Saudi Arabia"}]},{"given":"Saleh S.","family":"Baakeem","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Manufacturing Engineering, University of Calgary, 2500 University Drive, NW, Calgary, AB T2N 1N4, Canada"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4227-8468","authenticated-orcid":false,"given":"Abdulmajeed A.","family":"Mohamad","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Manufacturing Engineering, University of Calgary, 2500 University Drive, NW, Calgary, AB T2N 1N4, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s10444-004-1812-x","article-title":"Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids","volume":"23","author":"Fornberg","year":"2005","journal-title":"Adv. Comput. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Fasshauer, G.E. (2007). Meshfree Approximation Methods with Matlab: (With CD-ROM), World Scientific.","DOI":"10.1142\/6437"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1023\/A:1018932227617","article-title":"Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration","volume":"11","author":"Beatson","year":"1999","journal-title":"Adv. Comput. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1413","DOI":"10.1016\/j.mcm.2005.01.002","article-title":"Preconditioning for radial basis functions with domain decomposition methods","volume":"40","author":"Ling","year":"2004","journal-title":"Math. Comput. Model."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s10444-004-1809-5","article-title":"A least-squares preconditioner for radial basis functions collocation methods","volume":"23","author":"Ling","year":"2005","journal-title":"Adv. Comput. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.enganabound.2004.05.006","article-title":"On approximate cardinal preconditioning methods for solving PDEs with radial basis functions","volume":"29","author":"Brown","year":"2005","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Fasshauer, G.E., and Zhang, J.G. (2009). Preconditioning of radial basis function interpolation systems via accelerated iterated approximate moving least squares approximation. Progress on Meshless Methods, Springer.","DOI":"10.1007\/978-1-4020-8821-6_4"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/S0898-1221(01)00288-7","article-title":"Preconditioned conjugate gradients, radial basis functions, and Toeplitz matrices","volume":"43","author":"Baxter","year":"2002","journal-title":"Comput. Math. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1002\/nla.774","article-title":"Spectral analysis and preconditioning techniques for radial basis function collocation matrices","volume":"19","author":"Cavoretto","year":"2012","journal-title":"Numer. Linear Algebra Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.enganabound.2014.04.019","article-title":"Regularized symmetric positive definite matrix factorizations for linear systems arising from RBF interpolation and differentiation","volume":"44","author":"Sarra","year":"2014","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_11","unstructured":"Kansa, E., and Holoborodko, P. (2018, September 01). Strategies for Ill-Conditioned Radial Basis Functions Equations. Available online: https:\/\/www.researchgate.net\/profile\/Edward_Kansa\/publication\/320909429_Strategies_for_Ill-conditioned_radial_basis_functions_equations\/links\/5a01e5b0a6fdcc55a15816b9\/Strategies-for-Ill-conditioned-radial-basis-functions-equations.pdf."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.enganabound.2017.02.006","article-title":"On the ill-conditioned nature of C$\u0131nfty$ RBF strong collocation","volume":"78","author":"Kansa","year":"2017","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1016\/j.camwa.2003.08.010","article-title":"Stable computation of multiquadric interpolants for all values of the shape parameter","volume":"48","author":"Fornberg","year":"2004","journal-title":"Comput. Math. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1239","DOI":"10.1016\/j.enganabound.2009.07.003","article-title":"A random variable shape parameter strategy for radial basis function approximation methods","volume":"33","author":"Sarra","year":"2009","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Wang, S., Li, S., Huang, Q., and Li, K. (2012). An improved collocation meshless method based on the variable shaped radial basis function for the solution of the interior acoustic problems. Math. Probl. Eng., 2012.","DOI":"10.1155\/2012\/632072"},{"key":"ref_16","first-page":"260","article-title":"A new variable shape parameter strategy for Gaussian radial basis function approximation methods","volume":"42","author":"Ranjbar","year":"2015","journal-title":"Ann. Univ. Craiova-Math. Comput. Sci. Ser."},{"key":"ref_17","unstructured":"Golbabai, A., and Mohebianfar, E. (2018, September 01). A New Variable Shaped Radial Basis Function Approach for Solving European Option Pricing Model. Available online: http:\/\/oaji.net\/articles\/2015\/1719-1427235215.pdf."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1023\/A:1018975909870","article-title":"An algorithm for selecting a good value for the parameter c in radial basis function interpolation","volume":"11","author":"Rippa","year":"1999","journal-title":"Adv. Comput. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1016\/j.enganabound.2010.03.002","article-title":"On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs","volume":"34","author":"Huang","year":"2010","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2143","DOI":"10.1016\/j.camwa.2011.06.037","article-title":"On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation","volume":"62","author":"Davydov","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1137\/11S010840","article-title":"Choosing basis functions and shape parameters for radial basis function methods","volume":"4","author":"Mongillo","year":"2011","journal-title":"SIAM Undergrad. Res. Online"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.compstruct.2011.08.001","article-title":"A novel algorithm for shape parameter selection in radial basis functions collocation method","volume":"94","author":"Gherlone","year":"2012","journal-title":"Compos. Struct."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.enganabound.2014.10.018","article-title":"Laurent series based RBF-FD method to avoid ill-conditioning","volume":"52","author":"Bayona","year":"2015","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Biazar, J., and Hosami, M. (2016). Selection of an interval for variable shape parameter in approximation by radial basis functions. Adv. Numer. Anal., 2016.","DOI":"10.1155\/2016\/1397849"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.amc.2017.07.047","article-title":"An interval for the shape parameter in radial basis function approximation","volume":"315","author":"Biazar","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_26","first-page":"1","article-title":"Solving partial differential equations by collocation with radial basis functions","volume":"Volume 1997","author":"Fasshauer","year":"1996","journal-title":"Proceedings of Chamonix"},{"key":"ref_27","first-page":"23","article-title":"Stable PDE solution methods for large multiquadric shape parameters","volume":"25","author":"Libre","year":"2008","journal-title":"CMES Comput. Model. Eng. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1016\/j.cam.2011.06.030","article-title":"Better bases for radial basis function interpolation problems","volume":"236","author":"Beatson","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cam.2013.03.048","article-title":"A new stable basis for radial basis function interpolation","volume":"253","author":"Santin","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.camwa.2016.04.048","article-title":"A stable method for the evaluation of Gaussian radial basis function solutions of interpolation and collocation problems","volume":"72","author":"Rashidinia","year":"2016","journal-title":"Comput. Math. Appl."},{"key":"ref_31","unstructured":"Yurova, A., and Kormann, K. (2017). Stable evaluation of Gaussian radial basis functions using Hermite polynomials. arXiv."},{"key":"ref_32","first-page":"10","article-title":"RBF collocation methods as pseudospectral methods","volume":"39","author":"Fasshauer","year":"2005","journal-title":"WIT Trans. Model. Simul."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/s11075-007-9072-8","article-title":"On choosing \u201coptimal\u201d shape parameters for RBF approximation","volume":"45","author":"Fasshauer","year":"2007","journal-title":"Numer. Algorithms"},{"key":"ref_34","first-page":"207","article-title":"Using radial basis functions in a \u201cfinite difference mode\u201d","volume":"7","author":"Tolstykh","year":"2005","journal-title":"CMES Comput. Model. Eng. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.jcp.2005.05.030","article-title":"Scattered node compact finite difference-type formulas generated from radial basis functions","volume":"212","author":"Wright","year":"2006","journal-title":"J. Comput. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.jcp.2015.12.015","article-title":"Radial basis function interpolation in the limit of increasingly flat basis functions","volume":"307","author":"Kindelan","year":"2016","journal-title":"J. Comput. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.jcp.2016.05.026","article-title":"On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy","volume":"321","author":"Flyer","year":"2016","journal-title":"J. Comput. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.jcp.2016.12.008","article-title":"On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs","volume":"332","author":"Bayona","year":"2017","journal-title":"J. Comput. Phys."},{"key":"ref_39","unstructured":"Charles, C., Larry, L.S., and Joachim, S. (2002). Fast evaluation of radial basis functions: Methods based on partition of unity. The Approximation Theory X: Wavelets, Splines, and Applications, CiteseerX."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"070003","DOI":"10.1063\/1.4965349","article-title":"RBF-PU interpolation with variable subdomain sizes and shape parameters","volume":"Volume 1776","author":"Cavoretto","year":"2016","journal-title":"Proceedings of the AIP Conference Proceedings"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.apnum.2016.07.005","article-title":"Partition of unity interpolation using stable kernel-based techniques","volume":"116","author":"Cavoretto","year":"2017","journal-title":"Appl. Numer. Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1137\/060671991","article-title":"A stable algorithm for flat radial basis functions on a sphere","volume":"30","author":"Fornberg","year":"2007","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2758","DOI":"10.1016\/j.jcp.2007.11.016","article-title":"On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere","volume":"227","author":"Fornberg","year":"2008","journal-title":"J. Comput. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1137\/09076756X","article-title":"Stable Computations with Gaussian Radial Basis Functions","volume":"33","author":"Fornberg","year":"2011","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1137\/110824784","article-title":"Stable Evaluation of Gaussian Radial Basis Function Interpolants","volume":"34","year":"2012","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"A2096","DOI":"10.1137\/120899108","article-title":"Stable computation of differentiation matrices and scattered node stencils based on gaussian radial basis functions","volume":"35","author":"Larsson","year":"2013","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1016\/j.camwa.2012.11.006","article-title":"Stable calculation of Gaussian-based RBF-FD stencils","volume":"65","author":"Fornberg","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_48","first-page":"146","article-title":"Robust rational interpolation and least-squares","volume":"38","author":"Gonnet","year":"2011","journal-title":"Electron. Trans. Numer. Anal."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.jcp.2016.11.030","article-title":"Stable computations with flat radial basis functions using vector-valued rational approximations","volume":"331","author":"Wright","year":"2017","journal-title":"J. Comput. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1093\/imanum\/drt071","article-title":"Interpolation with variably scaled kernels","volume":"35","author":"Bozzini","year":"2014","journal-title":"IMA J. Numer. Anal."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Romani, L., Rossini, M., and Schenone, D. (2018). Edge detection methods based on RBF interpolation. J. Comput. Appl. Math., 349.","DOI":"10.1016\/j.cam.2018.08.006"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"De Marchi, S., Marchetti, F., and Perracchione, E. (2019). Jumping with variably scaled discontinuous kernels (VSDKs). BIT Numer. Math., 1\u201323.","DOI":"10.1007\/s10543-019-00786-z"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"B472","DOI":"10.1137\/19M1248777","article-title":"Shape-Driven Interpolation With Discontinuous Kernels: Error Analysis, Edge Extraction, and Applications in Magnetic Particle Imaging","volume":"42","author":"Erb","year":"2020","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Shepard, D. (1968). A Two-Dimensional Interpolation Function for Irregularly-Spaced Data. Proceedings of the 1968 23rd ACM National Conference, Association for Computing Machinery.","DOI":"10.1145\/800186.810616"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10915-017-0418-7","article-title":"Optimal Selection of Local Approximants in RBF-PU Interpolation","volume":"74","author":"Cavoretto","year":"2018","journal-title":"J. Sci. Comput."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/1\/1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:48:29Z","timestamp":1760179709000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/1\/1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,22]]},"references-count":55,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["a14010001"],"URL":"https:\/\/doi.org\/10.3390\/a14010001","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2020,12,22]]}}}