On the search of the shape parameter in radial basis functions using univariate global optimization methods

Cavoretto, Roberto; Rossi, Alessandra De; Mukhametzhanov, Marat S.; Sergeyev, Yaroslav D.

doi:10.1007/s10898-019-00853-3

Cited by 72 publications

(37 citation statements)

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“…They considered boundary data (as known solution) to mimic the actual error trend for selection of MQ shape parameter. One can also use the LOOCV (leave-oneout cross validation) algorithm and its modified versions for shape parameter selection, see Cavoretto et al (2019) for a detail discussion.…”

Section: Computational Results and Discussionmentioning

confidence: 99%

“…It is emphasized that the shape parameter c plays a vital role in controlling accuracy and stability. For relevant discussion on this aspect, see (Fasshauer 2007;Sarra and Kansa 2009;Fasshauer and McCourt 2015;Haq 2020a, 2019;Cavoretto et al 2019). In Table 1, note that the globally supported RBFs Sn and TPS RBFs belong to continuous functions class C 2n−1 .…”

Section: Hrbf Approximation Schemementioning

confidence: 99%

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Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers’ equation

Hussain

2021

Comp. Appl. Math.

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An efficient and accurate hybrid radial basis function (HRBF) method is proposed to numerically solve the quasi-linear viscous Burgers' equation. Two different RBFs are combined: an infinite smooth RBF defined by a shape parameter and a piecewise smooth RBF independent of the shape parameter. This combination formulates a family of HRBF methods, which is then used to approximate the spatial operator of the viscous Burgers equation. An efficient high-order solver is then used to integrate in time the resulting initial value problem. The developed numerical scheme is tested on some benchmark problems varying the Reynolds number. Accuracy and efficiency is validated using L ∞ , L 2 and L rms error norms, as well as the number of nodes N over the domain of influence. A rigorous comparison is conducted against well-known numerical schemes to manifest improved accuracy of the proposed scheme. Eigenvalue stability analysis of the proposed technique is thoroughly discussed and confirmed by numerical examples. The proposed scheme is found a well-behaved alternative to RBF (Kansa) and RBF-PS methods with improved accuracy and good conditioning properties. Keywords Burgers' equation• Hybrid RBF • Method of lines • Eigenvalues stability • Reynolds number • RBF-PS and Kansa method

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Section: Computational Results and Discussionmentioning

confidence: 99%

Section: Hrbf Approximation Schemementioning

confidence: 99%

Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers’ equation

Hussain

2021

Comp. Appl. Math.

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“…The linear system (5) has therefore a unique solution and provides a unique GBF approximation x * . For a vanishing regularization parameter γ → 0, the limit x • = lim γ →0 x * is uniquely determined by the condition (4) and the unique coefficients calculated in (5) with γ = 0. The resulting signal x • interpolates the data (w i , x(w i )), i.e.…”

Section: Positive Definite Gbfs For Signal Approximation On Graphsmentioning

confidence: 99%

“…Fig 5. RRMSEs and CPU times (in seconds) obtained for the GBF-PUM (with J = 8 and r = 8) and the global GBF scheme in terms of N interpolation nodes.…”

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confidence: 99%

Partition of Unity Methods for Signal Processing on Graphs

Cavoretto

Rossi

Erb

2021

J Fourier Anal Appl

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Partition of unity methods (PUMs) on graphs are simple and highly adaptive auxiliary tools for graph signal processing. Based on a greedy-type metric clustering and augmentation scheme, we show how a partition of unity can be generated in an efficient way on graphs. We investigate how PUMs can be combined with a local graph basis function (GBF) approximation method in order to obtain low-cost global interpolation or classification schemes. From a theoretical point of view, we study necessary prerequisites for the partition of unity such that global error estimates of the PUM follow from corresponding local ones. Finally, properties of the PUM as cost-efficiency and approximation accuracy are investigated numerically. KeywordsPartition of unity method (PUM) • Graph signal processing • Graph basis functions (GBFs) • Kernel-based approximation and interpolation • Spectral graph theory B Wolfgang Erb

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“…However, this may deceive one and gives a local minima instead of global one. This drawback has been recently dealt using global optimization tools [25]. Further investigations of Fasshaeur modification were carried out by Uddin [26] in solution of time‐dependent PDEs.…”

Section: Introductionmentioning

confidence: 99%

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

Hussain¹,

Haq

2020

Numerical Methods Partial

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In this work, we propose a hybrid radial basis functions (RBFs) collocation technique for the numerical solution of fractional advection-diffusion models. In the formulation of hybrid RBFs (HRBFs), there exist shape parameter (c*) and weight parameter () that control numerical accuracy and stability. For these parameters, an adaptive algorithm is developed and validated. The proposed HRBFs method is tested for numerical solutions of some fractional Black-Sholes and diffusion models. Numerical simulations performed for several benchmark problems verified the proposed method accuracy and efficiency. The quantitative analysis is made in terms of L ∞ , L 2 , L rms , and L rel error norms as well as number of nodes N over space domain and time-step t. Numerical convergence in space and time is also studied for the proposed method. The unconditional stability of the proposed HRBFs scheme is obtained using the von Neumann methodology. It is observed that the HRBFs method circumvented the ill-conditioning problem greatly, a major issue in the Kansa method.

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On the search of the shape parameter in radial basis functions using univariate global optimization methods

Cited by 72 publications

References 50 publications

Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers’ equation

Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers’ equation

Partition of Unity Methods for Signal Processing on Graphs

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

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