A numerical meshless method of soliton-like structures model via an optimal sampling density based kernel interpolation

Xu, Xingwang; Lü, Zhenzhou; Luo, Xiaopeng

doi:10.1016/j.cpc.2015.02.016

Cited by 4 publications

(3 citation statements)

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“…Multisymplectic methods [40,41], including a systematic method for discretizing Hamiltonian partial differential equations preserving their energy exactly [32,42], even for arbitrary boundary conditions [43]. Moreover, meshless methods based on multiquadric quasi-interpolation [44,45], on radial basis fuctions [46,47], and on an optimal nodal distribution determined by the so-called optimal sampling density of kernel interpolation time variables [48]. Even, exponentially-fitted and piecewise analytical methods [49], boundary element methods [50], local discontinuous Galerkin methods [51,52], and numerical implementations of the inverse scattering transform have been developed for the sGE [53,54].…”

Section: Introductionmentioning

confidence: 99%

Padé numerical schemes for the sine-Gordon equation

Martin-Vergara

Rus

Villatoro

2019

Applied Mathematics and Computation

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Section: Introductionmentioning

confidence: 99%

Padé numerical schemes for the sine-Gordon equation

Martin-Vergara

Rus

Villatoro

2019

Applied Mathematics and Computation

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“…In the last five decades, radial basis function (RBF) methods have gradually become an extremely powerful tool for scattered data. This is not only because they possess the dimensional independence and remarkable convergence properties (see, e.g., [1][2][3][4][5]), but also because a number of techniques, such as multipole (far-field) expansions [2,6], multilevel methods of compactly supported kernels [2,[7][8][9] and partition of unity methods [2,10,11], have been proposed to reduce both the condition number of the resulting interpolation matrix and the complexity of calculating the interpolant. These techniques are, of course, very important in practice, however, in contrast to the stability and efficiency, maybe the later question is the most crucial one for a general representation of functions, that is, how to accurately capture and represent the intrinsic structures of a target function, especially in high dimensional space.…”

Section: Introductionmentioning

confidence: 99%

Sparse residual tree and forest

Xu¹,

Luo²

2019

Preprint

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Sparse residual tree (SRT) is an adaptive exploration method for multivariate scattered data approximation. It leads to sparse and stable approximations in areas where the data is sufficient or redundant, and points out the possible local regions where data refinement is needed. Sparse residual forest (SRF) is a combination of SRT predictors to further improve the approximation accuracy and stability according to the error characteristics of SRTs. The hierarchical parallel SRT algorithm is based on both tree decomposition and adaptive radial basis function (RBF) explorations, whereby for each child a sparse and proper RBF refinement is added to the approximation by minimizing the norm of the residual inherited from its parent. The convergence results are established for both SRTs and SRFs. The worst case time complexity of SRTs is O(N log 2 N) for the initial work and O(log 2 N) for each prediction, meanwhile, the worst case storage requirement is O(N log 2 N), where the N data points can be arbitrary distributed. Numerical experiments are performed for several illustrative examples.

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“…A series of meshless approaches have been presented [4,12,31,32,35,47,48,52,58,62]. Moreover, since the nodal distribution for most existing meshless methods is preassigned, Xu et al [65] proposed a numerical two-step meshless method for soliton-like structures based on the optimal sampling density of kernel interpolation.…”

mentioning

confidence: 99%

Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

Luo

Rabitz

2019

J Math Chem

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In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs.Keywords Sine-Gordon equations · Meshless methods · High dimensional model representation 1 IntroductionThe sine-Gordon equation (SGE) is a nonlinear hyperbolic partial differential equation (PDE) involving the d'Alembert operator and the sine of the unknown function, and the SGE plays an important role in many mathematical physics applications. It was originally introduced by Bour [6] and rediscovered

show abstract

A numerical meshless method of soliton-like structures model via an optimal sampling density based kernel interpolation

Cited by 4 publications

References 64 publications

Padé numerical schemes for the sine-Gordon equation

Padé numerical schemes for the sine-Gordon equation

Sparse residual tree and forest

Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

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