{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T23:46:17Z","timestamp":1770335177638,"version":"3.49.0"},"reference-count":45,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T00:00:00Z","timestamp":1769904000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T00:00:00Z","timestamp":1769904000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2025,8,28]],"date-time":"2025-08-28T00:00:00Z","timestamp":1756339200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Applied Mathematics and Computation"],"published-print":{"date-parts":[[2026,2]]},"DOI":"10.1016\/j.amc.2025.129706","type":"journal-article","created":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T01:14:17Z","timestamp":1757380457000},"page":"129706","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":1,"special_numbering":"C","title":["Learning a robust shape parameter for RBF approximation"],"prefix":"10.1016","volume":"510","author":[{"given":"Maria","family":"Han Veiga","sequence":"first","affiliation":[]},{"given":"Faezeh","family":"Nassajian Mojarrad","sequence":"additional","affiliation":[]},{"given":"Fatemeh","family":"Nassajian Mojarrad","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/j.amc.2025.129706_bib0001","series-title":"Scattered Data Approximation","author":"Wendland","year":"2004"},{"issue":"9","key":"10.1016\/j.amc.2025.129706_bib0002","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/0898-1221(91)90123-L","article-title":"The parameter R2 in multiquadric interpolation","volume":"21","author":"Carlson","year":"1991","journal-title":"Comput. Math. Appl."},{"issue":"8","key":"10.1016\/j.amc.2025.129706_bib0003","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1029\/JB076i008p01905","article-title":"Multiquadric equations of topography and other irregular surfaces","volume":"76","author":"Hardy","year":"1971","journal-title":"J. Geophys. Res."},{"issue":"157","key":"10.1016\/j.amc.2025.129706_bib0004","first-page":"181","article-title":"Scattered data interpolation: tests of some methods","volume":"38","author":"Franke","year":"1982","journal-title":"Math. Comput."},{"key":"10.1016\/j.amc.2025.129706_bib0005","series-title":"Technical Report","article-title":"On the selection of a good shape parameter for RBF approximation and its application for solving PDEs","author":"Kuo","year":"2015"},{"issue":"8","key":"10.1016\/j.amc.2025.129706_bib0006","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1016\/0898-1221(87)90043-5","article-title":"Interpolation and approximation of 3-D and 4-D scattered data","volume":"13","author":"Foley","year":"1987","journal-title":"Comput. Math. Appl."},{"issue":"12","key":"10.1016\/j.amc.2025.129706_bib0007","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/0898-1221(92)90174-G","article-title":"Improved accuracy of multiquadric interpolation using variable shape parameters","volume":"24","author":"Kansa","year":"1992","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.amc.2025.129706_bib0008","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."},{"issue":"1","key":"10.1016\/j.amc.2025.129706_bib0009","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/S0955-7997(96)00033-1","article-title":"Improved multiquadric approximation for partial differential equations","volume":"18","author":"Golberg","year":"1996","journal-title":"Eng. Anal. Bound. Elem."},{"issue":"1","key":"10.1016\/j.amc.2025.129706_bib0010","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s10444-010-9146-3","article-title":"An alternative procedure for selecting a good value for the parameter c in RBF-interpolation","volume":"34","author":"Scheuerer","year":"2011","journal-title":"Adv. Comput. Math."},{"key":"10.1016\/j.amc.2025.129706_bib0011","doi-asserted-by":"crossref","DOI":"10.1016\/j.aml.2021.107262","article-title":"The extension of Rippa\u00e2s algorithm beyond LOOCV","volume":"120","author":"Marchetti","year":"2021","journal-title":"Appl. Math. Lett."},{"key":"10.1016\/j.amc.2025.129706_bib0012","doi-asserted-by":"crossref","DOI":"10.1016\/j.aml.2022.107955","article-title":"A stochastic extended Rippa\u00e2s algorithm for LpOCV","volume":"129","author":"Ling","year":"2022","journal-title":"Appl. Math. Lett."},{"issue":"3","key":"10.1016\/j.amc.2025.129706_bib0013","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/j.camwa.2007.01.028","article-title":"The Runge phenomenon and spatially variable shape parameters in, RBF interpolation","volume":"54","author":"Fornberg","year":"2007","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.amc.2025.129706_bib0014","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/s11075-007-9072-8","article-title":"On choosing \u00e2optimal\u00e2 shape parameters for RBF approximation","volume":"45","author":"Fasshauer","year":"2007","journal-title":"Numer. Algorithms"},{"issue":"4","key":"10.1016\/j.amc.2025.129706_bib0015","doi-asserted-by":"crossref","first-page":"3641","DOI":"10.1016\/j.aej.2017.11.024","article-title":"Finding a good shape parameter of RBF to solve PDEs based on the particle swarm optimization algorithm","volume":"57","author":"Koupaei","year":"2018","journal-title":"Alex. Eng. J."},{"issue":"8","key":"10.1016\/j.amc.2025.129706_bib0016","doi-asserted-by":"crossref","first-page":"1856","DOI":"10.3390\/math11081856","article-title":"Model for choosing the shape parameter in the multiquadratic radial basis function interpolation of an arbitrary sine wave and its application","volume":"11","author":"Sun","year":"2023","journal-title":"Mathematics"},{"key":"10.1016\/j.amc.2025.129706_bib0017","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2024.116108","article-title":"Parameter tuning in the radial kernel-based partition of unity method by Bayesian optimization","volume":"451","author":"Cavoretto","year":"2024","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.amc.2025.129706_bib0018","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2023.115716","article-title":"Bayesian approach for radial kernel parameter tuning","volume":"441","author":"Cavoretto","year":"2024","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"10.1016\/j.amc.2025.129706_bib0019","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s00365-022-09592-3","article-title":"Analysis of target data-dependent greedy kernel algorithms: convergence rates for f-, f.P- and f\/P-Greedy","volume":"57","author":"Wenzel","year":"2023","journal-title":"Constr. Approx."},{"key":"10.1016\/j.amc.2025.129706_bib0020","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2021.110378","article-title":"A greedy non-intrusive reduced order model for shallow water equations","volume":"439","author":"Dutta","year":"2021","journal-title":"J. Comput. Phys."},{"issue":"1","key":"10.1016\/j.amc.2025.129706_bib0021","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1137\/23M1551201","article-title":"Data-driven kernel designs for optimized greedy schemes: a machine learning perspective","volume":"46","author":"Wenzel","year":"2024","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.amc.2025.129706_bib0022","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/j.camwa.2023.05.005","article-title":"A new variable shape parameter strategy for RBF approximation using neural networks","volume":"143","author":"Mojarrad","year":"2023","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.amc.2025.129706_bib0023","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."},{"issue":"3","key":"10.1016\/j.amc.2025.129706_bib0024","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/BF02432002","article-title":"Error estimates and condition numbers for radial basis function interpolation","volume":"3","author":"Schaback","year":"1995","journal-title":"Adv. Comput. Math."},{"key":"10.1016\/j.amc.2025.129706_bib0025","series-title":"Meshfree Approximation Methods with MATLAB","author":"Fasshauer","year":"2007"},{"issue":"3\u20135","key":"10.1016\/j.amc.2025.129706_bib0026","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1016\/S0898-1221(01)00295-4","article-title":"Interpolation in the limit of increasingly flat radial basis functions","volume":"43","author":"Driscoll","year":"2002","journal-title":"Comput. Math. Appl."},{"issue":"1","key":"10.1016\/j.amc.2025.129706_bib0027","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1137\/19M129677X","article-title":"Spectral properties of kernel matrices in the flat limit","volume":"42","author":"Barthelm\u00e9","year":"2021","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"10.1016\/j.amc.2025.129706_bib0028","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1162\/neco.1992.4.4.473","article-title":"Simplifying neural networks by soft weight-sharing","volume":"4","author":"Nowlan","year":"1992","journal-title":"Neural Comput."},{"key":"10.1016\/j.amc.2025.129706_bib0029","unstructured":"D.P. Kingma, J. Ba, Adam: a method for stochastic optimization, arXiv: 1412.6980CoRR abs\/1412.6980(2014)."},{"issue":"2","key":"10.1016\/j.amc.2025.129706_bib0030","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1561\/2200000018","article-title":"Online learning and online convex optimization","volume":"4","author":"Shalev-Shwartz","year":"2012","journal-title":"Found. Trends-\u00aeMach. Learn."},{"issue":"13","key":"10.1016\/j.amc.2025.129706_bib0031","doi-asserted-by":"crossref","first-page":"3521","DOI":"10.1073\/pnas.1611835114","article-title":"Overcoming catastrophic forgetting in neural networks","volume":"114","author":"Kirkpatrick","year":"2017","journal-title":"Proc. Natl. Acad. Sci."},{"key":"10.1016\/j.amc.2025.129706_bib0032","unstructured":"M. Han Veiga, F.N. Mojarrad, Robust RBF shape parameter github, 2024. https:\/\/github.com\/hanveiga\/RBF-shape-parameter-NN."},{"issue":"3","key":"10.1016\/j.amc.2025.129706_bib0033","doi-asserted-by":"crossref","DOI":"10.1002\/amp2.10095","article-title":"The IDAES process modeling framework and model library\u2013Flexibility for process simulation and optimization","volume":"3","author":"Lee","year":"2021","journal-title":"J. Adv. Manuf. Process."},{"issue":"3\u20135","key":"10.1016\/j.amc.2025.129706_bib0034","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/S0898-1221(01)00296-6","article-title":"Newton iteration with multiquadrics for the solution of nonlinear PDEs","volume":"43","author":"Fasshauer","year":"2002","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.amc.2025.129706_bib0035","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1023\/A:1016609007255","article-title":"Newton iteration for partial differential equations and the approximation of the identity","volume":"25","author":"Fasshauer","year":"2000","journal-title":"Numer. Algorithms"},{"key":"10.1016\/j.amc.2025.129706_bib0036","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.enganabound.2017.11.012","article-title":"Improved Kansa RBF method for the solution of nonlinear boundary value problems","volume":"87","author":"Jankowska","year":"2018","journal-title":"Eng. Anal. Bound. Elem."},{"key":"10.1016\/j.amc.2025.129706_bib0037","series-title":"Digital Image Processing","author":"Gonzalez","year":"2002"},{"key":"10.1016\/j.amc.2025.129706_bib0038","first-page":"776","article-title":"Intelligent selection tools","volume":"2","author":"Mortensen","year":"2000","journal-title":"Proc. IEEE Conf. Computerv. Pattern Recognit."},{"issue":"1","key":"10.1016\/j.amc.2025.129706_bib0039","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1007\/s00466-003-0501-9","article-title":"On using radial basis functions in a \u00e2finite difference mode\u00e2 with applications to elasticity problems","volume":"33","author":"Tolstykh","year":"2003","journal-title":"Comput. Mech."},{"key":"10.1016\/j.amc.2025.129706_bib0040","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2021.110283","article-title":"An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries","volume":"436","author":"Tominec","year":"2021","journal-title":"J. Comput. Phys."},{"issue":"98","key":"10.1016\/j.amc.2025.129706_bib0041","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1090\/S0025-5718-1967-0225494-5","article-title":"The numerical integration of ordinary differential equations","volume":"21","author":"Gear","year":"1967","journal-title":"Math. Comput."},{"issue":"1","key":"10.1016\/j.amc.2025.129706_bib0042","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1007\/s10915-023-02427-8","article-title":"Energy-stable global radial basis function methods on summation-by-parts form","volume":"98","author":"Glaubitz","year":"2024","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.amc.2025.129706_bib0043","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2024.112889","article-title":"Summation-by-parts operators for general function spaces: the second derivative","volume":"504","author":"Glaubitz","year":"2024","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.amc.2025.129706_bib0044","series-title":"Interpolation of Spatial Statistics","author":"Stein","year":"1999"},{"key":"10.1016\/j.amc.2025.129706_bib0045","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s10898-019-00853-3","article-title":"On the search of the shape parameter in radial basis functions using univariate global optimization methods","volume":"79","author":"Cavoretto","year":"2021","journal-title":"J. Glob. Optim."}],"container-title":["Applied Mathematics and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0096300325004321?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0096300325004321?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T16:22:47Z","timestamp":1760804567000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0096300325004321"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2]]},"references-count":45,"alternative-id":["S0096300325004321"],"URL":"https:\/\/doi.org\/10.1016\/j.amc.2025.129706","relation":{},"ISSN":["0096-3003"],"issn-type":[{"value":"0096-3003","type":"print"}],"subject":[],"published":{"date-parts":[[2026,2]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Learning a robust shape parameter for RBF approximation","name":"articletitle","label":"Article Title"},{"value":"Applied Mathematics and Computation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.amc.2025.129706","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2025 The Author(s). Published by Elsevier Inc.","name":"copyright","label":"Copyright"}],"article-number":"129706"}}