A symmetric collocation method with fast evaluation

Johnson, Michael J.

doi:10.1093/imanum/drm046

Cited by 2 publications

(3 citation statements)

References 9 publications

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“…Over the years practical implementation of kernel approximation has progressed despite the ill-conditioning of kernel bases. This has happened with the help of clever numerical techniques like multipole methods and other fast methods of evaluation [1,4,12] and often with the help of preconditioners [3,6,14,23]. Many results already exist in the RBF literature concerning preconditioners and "better" bases.…”

Section: Introductionmentioning

confidence: 99%

“…A direct consequence of this is positive definiteness for such functions, |||u||| Ω,km ≥ 0 with equality only if u| Ω = 0. From this, we have a version of the Cauchy-Schwarz inequality: if u and v share a set of zeros Z (i.e., u| Z = v| Z = {0}) that is sufficiently dense in Ω, then | u, v Ω,km | ≤ |||u||| Ω,km |||v||| Ω,km (12) follows (sufficient density means that h(Z, Ω) < h * as above).…”

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confidence: 99%

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Localized Bases for Kernel Spaces on the Unit Sphere

Fuselier¹,

Hangelbroek²,

Narcowich³

et al. 2013

SIAM J. Numer. Anal.

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Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered data and is central to many meshless methods. For a set of N scattered sites, the standard basis for such a space utilizes N globally supported kernels; computing with it is prohibitively expensive for large N . Easily computable, well-localized bases with "small-footprint" basis elements-i.e., elements using only a small number of kernels-have been unavailable. Working on S 2 , with focus on the restricted surface spline kernels (e.g., the thin-plate splines restricted to the sphere), we construct easily computable, spatially well-localized, small-footprint, robust bases for the associated kernel spaces. Our theory predicts that each element of the local basis is constructed by using a combination of only O((log N ) 2 ) kernels, which makes the construction computationally cheap. We prove that the new basis is Lp stable and satisfies polynomial decay estimates that are stationary with respect to the density of the data sites, and we present a quasi-interpolation scheme that provides optimal Lp approximation orders. Although our focus is on S 2 , much of the theory applies to other manifolds-S d , the rotation group, and so on. Finally, we construct algorithms to implement these schemes and use them to conduct numerical experiments, which validate our theory for interpolation problems on S 2 involving over 150,000 data sites.

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Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

Localized Bases for Kernel Spaces on the Unit Sphere

Fuselier¹,

Hangelbroek²,

Narcowich³

et al. 2013

SIAM J. Numer. Anal.

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“…We propose an adaptive‐hybrid algorithm to try to overcome as many difficulties as possible. Using the freedom provided by meshfree methods not only allows for fast RBF evaluation as seen in the paper by Johnson , but also allows us to freely place data sites where most needed. In Section 2, an adaptive node refinement scheme is proposed to increase the density of data sites where most needed in regions with rapid variation.…”

Section: Introductionmentioning

confidence: 99%

An adaptive‐hybrid meshfree approximation method

Ling

2011

Numerical Meth Engineering

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SUMMARYIt is now commonly agreed that the global radial basis functions (GRBF) method is an attractive approach for approximating smooth functions. This superiority does not come free; one must find ways to circumvent the associated problem of ill-conditioning and the high computational cost for solving dense matrix systems. We previously proposed different variants of adaptive methods for selecting proper trial subspaces so that the instability caused by inappropriately shaped parameters were minimized. In contrast, the compactly supported radial basis functions (CSRBF) are more relaxing on the smoothness requirements. By settling with the algebraic order of convergence only, the CSRBF method, provided the support radii are properly chosen, can approximate functions with less smoothness. The reality is that end users must know the functions to be approximated a priori to decide which method to be used; this is not practical if one is solving a timeevolving partial differential equation. The solution could be smooth at the beginning but the formation of shocks may come later. In this paper, we propose a hybrid algorithm making use of both GRBF and CSRBF with other developed techniques for meshfree approximation with minimal fine tuning. The first contribution here is an adaptive node refinement scheme. Second, we apply the GRBFs (with adaptive subspace selection) on the adaptively generated data sites, and lastly, the CSRBF (with adaptive support selection) that can be used as a blackbox algorithm for robust approximations to a wider class of functions and for solving PDEs.

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A symmetric collocation method with fast evaluation

Cited by 2 publications

References 9 publications

Localized Bases for Kernel Spaces on the Unit Sphere

Localized Bases for Kernel Spaces on the Unit Sphere

An adaptive‐hybrid meshfree approximation method

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