On unsymmetric collocation by radial basis functions

Hon, Y.C.; Schaback, Robert

doi:10.1016/s0096-3003(99)00255-6

Cited by 189 publications

(143 citation statements)

References 12 publications

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“…Kansa [20] argued that if the centers of the RBFs are distinct and the PDE problem is wellposed, matrices discretizing spatial operators are generally found to be non-singular. Hon and Schaback [18] showed that occurrences of singular matrices are very rare, but do exist. When α(x, t) is constant, the mass matrix is constant over time.…”

Section: Fictitious Point Methodsmentioning

confidence: 99%

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

2017

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Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations because they are flexible with respect to the geometry of the computational domain, they can provide high order convergence, they are not more complicated for problems with many space dimensions and they allow for local refinement. The aim of this paper is to show that the solution of the Rosenau equation, as an example of an initial-boundary value problem with multiple boundary conditions, can be implemented using RBF approximation methods. We extend the fictitious point method and the resampling method to work in combination with an RBF collocation method. Both approaches are implemented in one and two space dimensions. The accuracy of the RBF fictitious point method is analyzed partly theoretically and partly numerically. The error estimates indicate that a high order of convergence can be achieved for the Rosenau equation. The numerical experiments show that both methods perform well. In the one-dimensional case, the accuracy of the RBF approaches is compared with that of the corresponding pseudospectral methods, showing similar or slightly better accuracy for the RBF methods. In the two-dimensional case, the Rosenau problem is solved both in a square domain and in an irregular domain with smooth boundary, to illustrate the capability of the RBF-based methods to handle irregular geometries.

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Section: Fictitious Point Methodsmentioning

confidence: 99%

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

2017

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“…Analysis of unsymmetric problems is hard, because even the solvability [12] of the finite subproblems is not evident.…”

Section: Symmetric Meshless Kernel Methodsmentioning

confidence: 99%

Solvability of partial differential equations by meshless kernel methods

Hon

Schaback²

2006

Adv Comput Math

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“…For each region, a radial basis method is applied to a much smaller number of points and the local interpolants are blended using C ′ rational hybrid cubic Bezier triangle functions. [126] gave a theoretical justi cation for combining the compactly supported RBF with numerical technique in space decomposition, which is based on Schwarz domain decomposition. The method overcomes the illconditioning problem resulted from using RBF as a global interpolant.…”

Section: Rbfns In Approximation and Interpolationmentioning

confidence: 99%

Radial basis function neural networks: a topical state-of-the-art survey

Dash¹,

Behera²,

Dehuri

et al. 2016

Open Computer Science

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Radial basis function networks (RBFNs) have gained widespread appeal amongst researchers and have shown good performance in a variety of application domains. They have potential for hybridization and demonstrate some interesting emergent behaviors. This paper aims to o er a compendious and sensible survey on RBF networks. The advantages they o er, such as fast training and global approximation capability with local responses, are attracting many researchers to use them in diversi ed elds. The overall algorithmic development of RBF networks by giving special focus on their learning methods, novel kernels, and ne tuning of kernel parameters have been discussed. In addition, we have considered the recent research work on optimization of multi-criterions in RBF networks and a range of indicative application areas along with some open source RBFN tools.

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On unsymmetric collocation by radial basis functions

Cited by 189 publications

References 12 publications

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

Solvability of partial differential equations by meshless kernel methods

Radial basis function neural networks: a topical state-of-the-art survey

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