On the new variable shape parameter strategies for radial basis functions

Golbabai, A.; Mohebianfar, Ehsan; Rabiei, Hamed

doi:10.1007/s40314-014-0132-0

Cited by 39 publications

(20 citation statements)

References 30 publications

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“…Ayrıca, sayısal yöntemin kararlılığı değerinden oldukça etkilenmektedir. Bu durum, daha önceki yapılan çalışmalarda belirtilmiştir [5,21]. Nokta sayısının artması ve değerinin azalması (fonksiyonun düz yatay çizgiye yaklaşması), matrisinin kötü koşullanmasına neden olarak Şekil 3'te görülen kararsızlıkları oluşturmaktadır.…”

Section: Poisson Problemi-1unclassified

Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon

Altınkaynak

2020

International Journal of Advances in Engineering and Pure Sciences

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Bu çalışmada, ağsız yöntemler için radyal özelliğe sahip yeni bir temel fonksiyon önerilmiştir. Önerilen fonksiyon, iki boyutta, dört farklı problemde, ağsız yöntemlerde sıklıkla kullanılan Ters Multikuadrik ve Gauss fonksiyonlarıyla birlikte test edilmiştir. Test problemlerinin üç tanesi 2. mertebeden mühendislik problemlerini içerirken son test problemi 4. mertebeden bir mühendislik problemi uygulaması olmuştur. 2. mertebeden test problemlerinde farklı sınır koşulları ve problem türleri incelenmiştir. Yapılan sayısal deneyler, önerilen fonksiyonun Ters Multikuadrik ve Gauss fonksiyonlarına kıyasla daha az nokta sayılarında benzer mertebedeki hatalara ulaşabildiğini göstermiştir. Ayrıca nokta sayısının artmasıyla aynı mertebedeki hatalar için kullanılabilecek şekil/ölçek parametresinin () diğer iki fonksiyona kıyasla daha geniş bir aralıkta seçilebildiği gösterilmiştir. Dolayısıyla, önerilen fonksiyon, ağsız yöntem uygulamalarında bir alternatif olarak kullanılabilecektir.

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Section: Poisson Problemi-1unclassified

Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon

Altınkaynak

2020

International Journal of Advances in Engineering and Pure Sciences

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“…Using variable shape parameters also helps reduce the Runge phenomenon in RBF interpolation . The search for good strategies for variable shape parameters is ongoing . This problem is analogous to the problem of selecting smoothing parameters in nonparametric curve estimation.…”

Section: System‐free Algorithm For Solving Pdesmentioning

confidence: 99%

“…29 The search for good strategies for variable shape parameters is ongoing. 30 This problem is analogous to the problem of selecting smoothing parameters in nonparametric curve estimation. A constant smoothing parameter, which typically is the minimizer of the mean integrated squared error estimated empirically by cross-validation or obtained theoretically by considering asymptotic expressions of bias and variance, is often good enough for the estimation of smoothed curves (see, eg, other works [31][32][33][34][35][36] ), but variable smoothing parameters will do a better job when the curves to be estimated have complicated structures, see other works.…”

Section: Quasi-random Shape Parametersmentioning

confidence: 99%

Fully adaptive kernel‐based methods

Ling

Chiu

2018

Numerical Meth Engineering

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Summary By exploiting the meshless property of kernel‐based collocation methods, we propose a fully automatic numerical recipe for solving interpolation/regression and boundary value problems adaptively. The proposed algorithm is built upon a least squares collocation formulation on some quasi‐random point sets with low discrepancy. A novel strategy is proposed to ensure that the fill distances of data points in the domain and on the boundary are in the same order of magnitude. To circumvent the potential problem of ill‐conditioning due to extremely small separation distance in the point sets, we add an extra dimension to the data points for generating shape parameters such that nearby kernels are of distinctive shape. This effectively eliminates the needs of shape parameter identification. Resulting linear systems were then solved by a greedy trial space algorithm to improve the robustness of the algorithm. Numerical examples are provided to demonstrate the efficiency and accuracy of the proposed methods.

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“…This is despite the fact that ORBIT uses a constant value as a shape parameter regardless of the optimization problem. There are several methods to determine a good shape parameter (Hardy 1971;Franke 1982;Golbabai et al 2015). In addition, many systematic approaches have been suggested in the statistics literature, see, for example (Wahba 1990).…”

Section: Introductionmentioning

confidence: 99%

An improved hybrid-ORBIT algorithm based on point sorting and MLE technique

et al. 2019

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Optimization using radial basis functions as an interpolation tool in trust region (ORBIT) is a derivative-free framework based on fully linear radial basis function (RBF) models. In this paper, an improved version of ORBIT algorithm based on two novel ideas is proposed. The accuracy and stability of RBFs depend on a so-called shape parameter, so it is more appropriate to determine the shape parameter according to the optimization problem. While ORBIT in all problems uses a fixed value as a shape parameter, our new version, Hybrid-ORBIT, uses a statistical technique to select an appropriate shape parameter. In addition, ORBIT uses some stored points to build a fully linear RBF model without considering their function values, while in the Hybrid-ORBIT algorithm the stored points are sorted based on their function values and the RBF model is built using the points with lower function values, and the best point in the sense of function value is defined as the trust-region center. Numerical results indicate the efficiency of the improved version compared with the original version.

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On the new variable shape parameter strategies for radial basis functions

Cited by 39 publications

References 30 publications

Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon

Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon

Fully adaptive kernel‐based methods

An improved hybrid-ORBIT algorithm based on point sorting and MLE technique

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