Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

Narcowich, Francis J.; Ward, Joseph D.; Wendland, Holger

doi:10.1090/s0025-5718-04-01708-9

Cited by 129 publications

(137 citation statements)

References 19 publications

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“…has a negative orbital derivative. Such error estimates were derived, for example in (Giesl, 2007;Giesl and Wendland, 2007), see also (Narcowich et al, 2005;Wendland, 2005).…”

Section: Mesh-free Collocationmentioning

confidence: 99%

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Argáez

Hafstein

Giesl

2017

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

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Abstract:Ordinary differential equations arise in a variety of applications, including e.g. climate systems, and can exhibit complicated dynamical behaviour. Complete Lyapunov functions can capture this behaviour by dividing the phase space into the chain-recurrent set, determining the long-time behaviour, and the transient part, where solutions pass through. In this paper, we present an algorithm to construct complete Lyapunov functions. It is based on mesh-free numerical approximation and uses the failure of convergence in certain areas to determine the chain-recurrent set. The algorithm is applied to three examples and is able to determine attractors and repellers, including periodic orbits and homoclinic orbits.

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“…has a negative orbital derivative. Such error estimates were derived, for example in (Giesl, 2007;Giesl and Wendland, 2007), see also (Narcowich et al, 2005;Wendland, 2005).…”

Section: Mesh-free Collocationmentioning

confidence: 99%

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Argáez

Hafstein

Giesl

2017

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

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“…It is a problem of Numerical Analysis to show that certain algorithms actually produce such approximants. Standard examples are in [21,19,28,29]. Here, we are satisfied with pointing out that such approximants exist under very weak conditions.…”

Section: Overcoming Low Regularitymentioning

confidence: 74%

Solvability of partial differential equations by meshless kernel methods

Hon

Schaback²

2006

Adv Comput Math

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“…However, if the native space is Sobolev one can instead use recent Sobolev estimates for functions with scattered zeros on R n . Specifically, one can use the following proposition, whose proof can be found in [29].…”

Section: Interpolation Error For Smooth and Rough Target Functionsmentioning

confidence: 99%

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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Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

Cited by 129 publications

References 19 publications

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Solvability of partial differential equations by meshless kernel methods

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

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