Laurent series based RBF-FD method to avoid ill-conditioning

González-Rodríguez, Pedro; Bayona, Víctor; Moscoso, Miguel; Kindelán, Manuel

doi:10.1016/j.enganabound.2014.10.018

Cited by 23 publications

(18 citation statements)

References 24 publications

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“…Because of the reluctance to utilize extended precision software packages and rather use single-or double-precision computers, alternatives to direct RBF methods were developed. Local RBF alternatives to direct CD-RBFs such as RBF-FD, RBF-QR, RBF-RA and Hermite-based RBFs are among the approaches that were developed [35][36][37][38][39][40]. Instead of spectral convergence rates, the convergence rates are fourth to sixth order.…”

Section: Local Rbf Methodsmentioning

confidence: 99%

Fully and sparsely supported radial basis functions

Kansa¹,

Holoborodko²

2020

Int. J. CMEM

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The central idea of this paper is that computer mathematics is not identical to ideal mathematics because computer numbers only have finite precision. All functions, especially the positive definite transcendental functions, are truncated and the expansion coefficients have finite precision and all branching operations require time to complete. Of all the known methods used to obtain numerical solutions to integral and partial differential equations, the global continuously differential radial basis functions (RBFs) that are implemented on computers closely resemble many aspects of ideal mathematics. The global RBFs have the attributes required to obtain very accurate numerical results for a variety of partial differential and integral equations with smooth solutions. Without the need for extremely fine discretization, the global RBFs have their spline properties and exponential convergence rates. The resulting system of full equations can be executed very rapidly on graphical processing units and fieldprogrammable gate arrays because, with full systems, there is no branching and full systems solvers are very highly vectorized, optimizing the usage of very fast processors.

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Section: Local Rbf Methodsmentioning

confidence: 99%

Fully and sparsely supported radial basis functions

Kansa¹,

Holoborodko²

2020

Int. J. CMEM

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“…In this example, we consider six cases with different shapes of fictitious sources using (19) to (24) as demonstrated in Figure 2a-f for Cases I, II, III, IV, V, and VI, respectively. The fictitious sources were placed on the external domain by utilizing (25). The fictitious source boundaries were defined as [33,34] ∂Ω s = (x s j , y s j ) x s j = ηρ s j cos θ s j , y s j = ηρ s j sin θ s j (25) where ∂Ω s denotes the boundaries of fictitious sources.…”

Section: Convergence Analysismentioning

confidence: 99%

“…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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“…To overcome this problem, the interpolation coefficients in the limit ϵ → 0 are computed in [18,23] by means of the Laurent series of the inverse of the interpolation matrix A.…”

Section: Rbf Interpolationmentioning

confidence: 99%

A Closed-Form Formula for the RBF-Based Approximation of the Laplace–Beltrami Operator

2018

Self Cite

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In this paper we present a method that uses radial basis functions to approximate the Laplace-Beltrami operator that allows to solve numerically diffusion (and reactiondiffusion) equations on smooth, closed surfaces embedded in R 3 . The novelty of the method is in a closed-form formula for the Laplace-Beltrami operator derived in the paper, which involve the normal vector and the curvature at a set of points on the surface of interest. An advantage of the proposed method is that it does not rely on the explicit knowledge of the surface, which can be simply defined by a set of scattered nodes. In that case, the surface is represented by a level set function from which we can compute the needed normal vectors and the curvature. The formula for the Laplace-Beltrami operator is exact for radial basis functions and it also depends on the first and second derivatives of these functions at the scattered nodes that define the surface. We analyze the converge of the method and we present numerical simulations that show its performance. We include an application that arises in cardiology.

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Laurent series based RBF-FD method to avoid ill-conditioning

Cited by 23 publications

References 24 publications

Fully and sparsely supported radial basis functions

Fully and sparsely supported radial basis functions

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

A Closed-Form Formula for the RBF-Based Approximation of the Laplace–Beltrami Operator

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