Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants

Fuselier, Edward J.

doi:10.1090/s0025-5718-07-02096-0

Cited by 29 publications

(30 citation statements)

References 23 publications

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“…The function Φ curl is positive definite and gives curl-free interpolants on R 3 [10,11]. In what follows we will rely on results already known for Φ curl to aid in our discussion of Ψ div , Ψ curl and Ψ.…”

Section: Constructing Kernels On S 2 From Rbfs In Rmentioning

confidence: 99%

“…The intial escape was first proven in the case of scalar RBFs by Narcowich, Ward and Wendland, and their technique has since been applied to various situations involving RBFs [11,12,26,27,30]. A common theme in all these cases is to use functions that are band-limited, that is, functions whose Fourier transforms are compactly supported.…”

Section: Interpolation and Approximation Via Vector Spherical Polynommentioning

confidence: 99%

“…As mentioned before, the kernel Φ curl , which gives curl-free interpolants in R 3 , has been studied: it is positive definite on R 3 and estimates similar to (16) have been given [11]. We will use (17) to transfer the problem from N Ψ to N Φ curl .…”

Section: Theorem 2 Let τ > 1 There Exists Constants κ and C Both Omentioning

confidence: 99%

“…Many of the arguments that follow have been applied in several other situations. Sobolev error estimates have been derived in a similar way for scalar and matrix-valued RBFs and scalar SBFs [11,26,27,30]. In fact, the methods we employ here have been recently used to derive similar results for the divergence-free tangent kernel Ψ div [12,Section 4].…”

Section: Interpolation Error For Smooth and Rough Target Functionsmentioning

confidence: 99%

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Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

Fuselier¹,

Wright²

2009

SIAM J. Numer. Anal.

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A new numerical technique based on radial basis functions (RBFs) is presented for fitting a vector field tangent to the sphere, S 2 , from samples of the field at "scattered" locations on S 2 . The method naturally provides a way to decompose the reconstructed field into its individual Helmholtz-Hodge components, i.e. into divergence-free and curl-free parts, which is useful in many applications from the atmospheric and oceanic sciences (e.g., in diagnosing the horizontal wind and ocean currents). Several approximation results for the method will be derived. In particular, Sobolev-type error estimates are obtained for both the interpolant and its decomposition. Optimal stability estimates for the associated interpolation matrices are also presented. Finally, numerical validation of the theoretical results is given for vector fields with similar characteristics to those of atmospheric wind fields.

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Section: Constructing Kernels On S 2 From Rbfs In Rmentioning

confidence: 99%

Section: Interpolation and Approximation Via Vector Spherical Polynommentioning

confidence: 99%

Section: Theorem 2 Let τ > 1 There Exists Constants κ and C Both Omentioning

confidence: 99%

Section: Interpolation Error For Smooth and Rough Target Functionsmentioning

confidence: 99%

See 2 more Smart Citations

Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

Fuselier¹,

Wright²

2009

SIAM J. Numer. Anal.

Self Cite

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“…The kernel Ψ curl := −∇∇ T ψ is the negative of the 3D Hessian of ψ and is a 3 × 3 matrixvalued RBF whose columns are curl free [29,8]. The kernel Ψ(x, y) takes vectors tangent to P at y and outputs vectors tangent at x.…”

Section: Introductionmentioning

confidence: 99%

Error and stability estimates for surface-divergence free RBF interpolants on the sphere

et al. 2009

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Abstract. Recently, a new class of surface-divergence free radial basis function interpolants has been developed for surfaces in R 3 . In this paper, several approximation results for this class of interpolants will be derived in the case of the sphere, S 2 . In particular, Sobolev-type error estimates are obtained, as well as optimal stability estimates for the associated interpolation matrices. In addition, a Bernstein estimate and an inverse theorem are also derived. Numerical validation of the theoretical results is also given.

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Matrix‐valued radial basis functions for the Lamé system

Grzhibovskis

Michel

Rjasanow

2018

Math Methods in App Sciences

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The problem of finding an approximate particular solution of an elliptic system of partial differential equations is considered. To construct the approximation, the differential operator is applied to a vector of radial basis functions. The resulting vectors are linearly combined to interpolate the function on the right hand side. The solvability of the interpolation problem is established. Additionally, stability, and accuracy estimates for the method are given. These theoretical results are illustrated on several numerical examples related to the Lamé system. KEYWORDSLamé system, matrix-valued RBF, particular solutions 6080

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Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants

Cited by 29 publications

References 23 publications

Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

Error and stability estimates for surface-divergence free RBF interpolants on the sphere

Matrix‐valued radial basis functions for the Lamé system

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