Fast and stable rational RBF-based partition of unity interpolation

Marchi, Stefano De; Martínez, Ángeles; Perracchione, Emma

doi:10.1016/j.cam.2018.07.020

Cited by 32 publications

(15 citation statements)

References 28 publications

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“…We stress that the current study might be useful for image reconstruction in the context of Magnetic Particle Imaging [6,7]. Finally, for smooth RBFs, we should study the behaviour of VSDKs when rational RBF interpolants are used [9,19].…”

Section: Discussionmentioning

confidence: 99%

Jumping with variably scaled discontinuous kernels (VSDKs)

2019

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In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernelbased basis functions. The linear spaces spanned by these discontinuous kernels lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon in reconstructing functions with jumps. For the new basis we provide error bounds and numerical results that support our claims. The method is also effectively tested for approximating satellite images.

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Section: Discussionmentioning

confidence: 99%

Jumping with variably scaled discontinuous kernels (VSDKs)

2019

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“…The versatility of scattered data interpolation techniques is confirmed by a lot of applications, e.g., surface reconstruction, image restoration and inpainting, meshless/Lagrangian methods for fluid dynamics, surface deformation or motion capture systems allowing the recording of sparse motions from deformable objects such as human faces and bodies [6]. The numerical solution of partial differential equations by a global collocation approach based on RBF, is also referred to as a strong form solution in the PDE literature [7][8][9][10]. An alternative interesting approach in collocation methods is to use other bases as for example Hermite exponential spline defined in [11].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Localized Meshless Method Based on the Space–Time Gaussian RBF for High-Dimensional Space Fractional Wave and Damped Equations

Raei

Cuomo

2021

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In this paper, an efficient localized meshless method based on the space–time Gaussian radial basis functions is discussed. We aim to deal with the left Riemann–Liouville space fractional derivative wave and damped wave equation in high-dimensional space. These significant problems as anomalous models could arise in several research fields of science, engineering, and technology. Since an explicit solution to such equations often does not exist, the numerical approach to solve this problem is fascinating. We propose a novel scheme using the space–time radial basis function with advantages in time discretization. Moreover this approach produces the (n + 1)-dimensional spatial-temporal computational domain for n-dimensional problems. Therefore the local feature, as a remarkable and efficient property, leads to a sparse coefficient matrix, which could reduce the computational costs in high-dimensional problems. Some benchmark problems for wave models, both wave and damped, have been considered, highlighting the proposed method performances in terms of accuracy, efficiency, and speed-up. The obtained experimental results show the computational capabilities and advantages of the presented algorithm.

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“…Despite the great success of the above RBFs as effective numerical techniques for dealing with several kinds of PDEs, there is still growing interest in the application and development of new and advanced RBFs [20]. A significant number of modifications to RBFs have been proposed, such as the pseudo-spectral RBF [21,22], Gaussian RBF [23], RBF QR alternative basis method [24], finite difference RBF [25,26], partition of unity RBF [27,28], stabilized expansion of the Gaussian RBF [29], rational RBF [30,31], and RBF based on partition of unity of Taylor series expansion [32].…”

Section: Introductionmentioning

confidence: 99%

Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs

Liu

Hong

et al. 2021

Mathematics

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In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required.

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Fast and stable rational RBF-based partition of unity interpolation

Cited by 32 publications

References 28 publications

Jumping with variably scaled discontinuous kernels (VSDKs)

Jumping with variably scaled discontinuous kernels (VSDKs)

An Efficient Localized Meshless Method Based on the Space–Time Gaussian RBF for High-Dimensional Space Fractional Wave and Damped Equations

Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs

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