Solving PDEs with radial basis functions

Fornberg, Bengt; Flyer, Natasha

doi:10.1017/s0962492914000130

Cited by 250 publications

(147 citation statements)

References 100 publications

(117 reference statements)

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“…Problems on spherical shells naturally arise in a wide range of geosciences, and numerous computational methods have been proposed (Flyer et al 2013;Fornberg and Flyer 2015;McWilliams 1996;Sakuraba and Kono 1999;Tackley 2008). Nonetheless, theoretical analysis does not seem to have attracted much attention.…”

Section: Introductionmentioning

confidence: 99%

A fully discretised polynomial approximation on spherical shells

Kazashi

2016

Int J Geomath

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A fully discretised polynomial approximation on spherical shells is considered. The radial direction and the angular direction of the spherical shells are treated differently: for the radial direction, polynomial interpolation is employed, whereas for the angular direction, hyperinterpolation is employed. The error in terms of a weighted L 2 -norm is bounded by the sum of two terms, one for the radial, one for the angular direction. Numerical results support our result.

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Section: Introductionmentioning

confidence: 99%

A fully discretised polynomial approximation on spherical shells

Kazashi

2016

Int J Geomath

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“…so that the constraint n i=1 w i = 0 can be satisfied and the solution exactly reproduces a constant (Lehto, 2012;Flyer et al, 2012Flyer et al, , 2015bFornberg and Flyer, 2015b, a). This results in a less oscillatory interpolant and thus more accurate derivative approximations.…”

Section: Rbf-fd Methodsmentioning

confidence: 96%

“…Some examples of smooth RBFs are listed in Table 1. Unlike FD, in which the interpolation problem is not guaranteed to be non-singular for scattered nodes in n dimensions (n ≥ 2), RBF-FD is guaranteed to be non-singular no matter how the n nodes (assumed distinct) are scattered in any number of dimensions (Fasshauer, 2007;Fornberg and Flyer, 2015b). Augmenting RBF interpolants with polynomials can be beneficial.…”

Section: Rbf-fd Methodsmentioning

confidence: 99%

“…Some of the main features of RBF-FD (Bayona et al, 2010;Bayona and Kindelan, 2013;Stevens et al, 2009;Lehto, 2012;Flyer et al, 2012Flyer et al, , 2015bFornberg and Flyer, 2015b, a) have proven to be very beneficial in modeling the GEC. For instance, RBF-FD is a meshless local method that only depends on the Euclidean distance between neighboring nodes.…”

Section: Rbf-fd Methodsmentioning

confidence: 99%

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A 3-D RBF-FD solver for modeling the atmospheric global electric circuit with topography (GEC-RBFFD v1.0)

et al. 2015

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Abstract. A numerical model based on radial basis functiongenerated finite differences (RBF-FD) is developed for simulating the global electric circuit (GEC) within the Earth's atmosphere, represented by a 3-D variable coefficient linear elliptic partial differential equation (PDE) in a spherically shaped volume with the lower boundary being the Earth's topography and the upper boundary a sphere at 60 km. To our knowledge, this is (1) the first numerical model of the GEC to combine the Earth's topography with directly approximating the differential operators in 3-D space and, related to this, (2) the first RBF-FD method to use irregular 3-D stencils for discretization to handle the topography. It benefits from the mesh-free nature of RBF-FD, which is especially suitable for modeling high-dimensional problems with irregular boundaries. The RBF-FD elliptic solver proposed here makes no limiting assumptions on the spatial variability of the coefficients in the PDE (i.e., the conductivity profile), the right hand side forcing term of the PDE (i.e., distribution of current sources) or the geometry of the lower boundary.

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“…They have been successfully used in various applications such as geography, engineering, neural networks, machine learning and image processing. Further applications include solving Partial Differential Equations (Fornberg and Flyer, 2015), the construction of Lyapunov functions in dynamical systems (Giesl, 2007), and high-dimensional integration (Dick et al, 2013).…”

Section: Introductionmentioning

confidence: 99%

Wendland Functions - A C++ Code to Compute Them

Argáez

Hafstein

Giesl

2017

Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

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Abstract:In this paper we present a code in C++ to compute Wendland functions for arbitrary smoothness parameters. Wendland functions are compactly supported Radial Basis Functions that are used for interpolation of data or solving Partial Differential Equations with mesh-free collocation. For the computations of Lyapunov functions using Wendland functions their derivatives are also needed so we include this in the code. Wendland functions with a few fixed smoothness parameters are included in some C++ libraries, but for the general case the only code freely available was implemented in MAPLE taking advantage of the computer algebra system. The aim of this contribution is to allow scientists to use Wendland functions in their C++ code without having to implement them themselves. The computed Wendland functions are polynomials and their coefficients are computed and stored in a vector, which allows for efficient computation of their values using the Horner scheme.

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Solving PDEs with radial basis functions

Cited by 250 publications

References 100 publications

A fully discretised polynomial approximation on spherical shells

A fully discretised polynomial approximation on spherical shells

A 3-D RBF-FD solver for modeling the atmospheric global electric circuit with topography (GEC-RBFFD v1.0)

Wendland Functions - A C++ Code to Compute Them

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