Localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for nonlocal diffusion problems

Zhao, Wei; Hon, Y.C.; Stoll, Martin

doi:10.1016/j.camwa.2017.11.030

Cited by 11 publications

(3 citation statements)

References 39 publications

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“…However, for large-scale problems such as image processing, the RBF method incurs excessive computational costs due to the generation of a dense matrix [23][24][25][26], and the large condition number of this matrix can also causes calculation instability [27,28]. To conquer this shortcoming, local RBF (LRBF) methods based on positive and conditionally positive global kernels have been developed, such as [29], which consider only the contributions from several neighboring points in the near field while ignoring the influence of distant points. The corresponding sparse interpolation matrix apparently reduces the condition number of the matrix, saves storage space and enhances computational efficiency.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Local Radial Basis Function Method for Image Segmentation Based on the Chan–Vese Model

Qiu,

Lin,

Zhao

2024

CMES

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In this paper, we consider the Chan-Vese (C-V) model for image segmentation and obtain its numerical solution accurately and efficiently. For this purpose, we present a local radial basis function method based on a Gaussian kernel (GA-LRBF) for spatial discretization. Compared to the standard radial basis function method, this approach consumes less CPU time and maintains good stability because it uses only a small subset of points in the whole computational domain. Additionally, since the Gaussian function has the property of dimensional separation, the GA-LRBF method is suitable for dealing with isotropic images. Finally, a numerical scheme that couples GA-LRBF with the fourth-order Runge-Kutta method is applied to the C-V model, and a comparison of some numerical results demonstrates that this scheme achieves much more reliable image segmentation.

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Section: Introductionmentioning

confidence: 99%

An Efficient Local Radial Basis Function Method for Image Segmentation Based on the Chan–Vese Model

Qiu,

Lin,

Zhao

2024

CMES

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“…The application of RBF-based methods to solve fractional PDEs and nonlocal problems is still very recent. In [5,34,49], the Galerkin methods using a localized basis of RBFs were proposed to solve nonlocal diffusion problems. In [38], RBF-QR methods were proposed to solve the Riemann-Liouville spatial fractional diffusion problems.…”

Section: Introductionmentioning

confidence: 99%

“…In this work, we propose a novel meshfree pseudospectral method based on the Gaussian RBFs, which has fundamental differences from other RBF-based methods in [5,34,37,38,40,46,49]. Inheriting the advantages of RBF methods, our method is simple and flexible of domain geometry, and its computer implementation remains the same for any dimension d ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

A unified meshfree pseudospectral method for solving both classical and fractional PDEs

Burkardt,

Wu,

Zhang

2020

Preprint

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In this paper, we propose a meshfree method based on the Gaussian radial basis function (RBF) to solve both classical and fractional PDEs. The proposed method takes advantage of the analytical Laplacian of Gaussian functions so as to accommodate the discretization of the classical and fractional Laplacian in a single framework and avoid the large computational cost for numerical evaluation of the fractional derivatives. These important merits distinguish it from other numerical methods for fractional PDEs. Moreover, our method is simple and easy to handle complex geometry and local refinement, and its computer program implementation remains the same for any dimension d ≥ 1. Extensive numerical experiments are provided to study the performance of our method in both approximating the Dirichlet Laplace operators and solving PDE problems. Compared to the recently proposed Wendland RBF method, our method exactly incorporates the Dirichlet boundary conditions into the scheme and is free of the Gibbs phenomenon as observed in the literature. Our studies suggest that to obtain good accuracy the shape parameter cannot be too small or too big, and the optimal shape parameter might depend on the RBF center points and the solution properties.

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Stability and convergence of a new hybrid method for fractional partial differential equations

2023

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Localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for nonlocal diffusion problems

Cited by 11 publications

References 39 publications

An Efficient Local Radial Basis Function Method for Image Segmentation Based on the Chan–Vese Model

An Efficient Local Radial Basis Function Method for Image Segmentation Based on the Chan–Vese Model

A unified meshfree pseudospectral method for solving both classical and fractional PDEs

Stability and convergence of a new hybrid method for fractional partial differential equations

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