Compactly Supported, Piecewise Polyharmonic Radial Functions with Prescribed Regularity

Johnson, Michael J.

doi:10.1007/s00365-011-9141-z

Cited by 9 publications

(13 citation statements)

References 16 publications

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“…A remarkable example of a Beppo Levi L 0 -spline has recently been obtained independently by Johnson [15] in the context of radial basis function (RBF) methods for multivariable scattered data interpolation. Specifically, the profile denoted by Johnson as η 2 , constructed as part of a family of compactly supported and piecewise polyharmonic RBFs, can be expressed as…”

Section: Two Compactly Supported L 0 -Splinesmentioning

confidence: 99%

“…We also prove that the exponent 3/2 in the above approximation order cannot be increased in general for data functions from the radial energy space. In section 5, we illustrate the numerical accuracy, as well as several graphs of the resulting interpolatory L-spline profiles, and point out a connection with Johnson's recent construction of compactly supported radial basis functions [15].…”

Section: Introductionmentioning

confidence: 99%

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Radially symmetric thin plate splines interpolating a circular contour map

Bejancu

2016

Journal of Computational and Applied Mathematics

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MSC:Keywords: Thin plate spline Radially symmetric function L-spline Beppo Levi polyspline Approximation order a b s t r a c t Profiles of radially symmetric thin plate spline surfaces minimizing the Beppo Levi energy over a compact annulus R 1 ≤ r ≤ R 2 have been studied by Rabut via reproducing kernel methods. Motivated by our recent construction of Beppo Levi polyspline surfaces, we focus here on minimizing the radial energy over the full semi-axis 0 < r < ∞. Using a L-spline approach, we find two types of minimizing profiles: one is the limit of Rabut's solution as R 1 → 0 and R 2 → ∞ (identified as a 'non-singular' L-spline), the other has a secondderivative singularity and matches an extra data value at 0. For both profiles and p ∈ [2, ∞], we establish the L p -approximation order 3/2 + 1/p in the radial energy space. We also include numerical examples and obtain a novel representation of the minimizers in terms of dilates of a basis function.

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Section: Two Compactly Supported L 0 -Splinesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Radially symmetric thin plate splines interpolating a circular contour map

Bejancu

2016

Journal of Computational and Applied Mathematics

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“…This was necessary at that time. Since then, a series of new compactly supported radial basis functions have emerged, so that, for example, it is now known that every H σ (R d ) with σ ∈ N, σ > d/2 has a compactly supported reproducing kernel, see [13,21].…”

Section: Stability Resultsmentioning

confidence: 99%

RBF Multiscale Collocation for Second Order Elliptic Boundary Value Problems

Farrell¹,

Wendland²

2013

SIAM J. Numer. Anal.

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In this paper, we discuss multiscale radial basis function collocation methods for solving elliptic partial differential equations on bounded domains. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. On each level, standard symmetric collocation is employed. A convergence theory is given, which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions. We are able to show that the convergence is linear in the number of levels. We also discuss the condition numbers of the arising systems and the effect of simple, diagonal preconditioners, now proving rigorously previous numerical observations.

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“…We will mostly be interested in radial basis functions with compact support having a Fourier transform with such a decay. Typical examples are given in Johnson (2012); Wendland (1995Wendland ( , 2005.…”

Section: Positive Definite Matrix-valued Kernelsmentioning

confidence: 99%

Multilevel interpolation of divergence-free vector fields

2016

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We introduce a multilevel technique for interpolating scattered data of divergence-free vector fields with the help of matrix-valued compactly supported kernels. The support radius at a given level is linked to the mesh norm of the data set at that level. There are at least three advantages of this method: no grid structure is necessary for the implementation, the multilevel approach is computationally cheaper than solving a large one-shot system and the interpolant is guaranteed to be analytically divergence-free. Furthermore, though we will not pursue this here, our multiscale approach is able to represent multiple scales in the data if present. We will prove convergence of the scheme, stability estimates and give a numerical example.

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Compactly Supported, Piecewise Polyharmonic Radial Functions with Prescribed Regularity

Cited by 9 publications

References 16 publications

Radially symmetric thin plate splines interpolating a circular contour map

Radially symmetric thin plate splines interpolating a circular contour map

RBF Multiscale Collocation for Second Order Elliptic Boundary Value Problems

Multilevel interpolation of divergence-free vector fields

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