Radial functions on compact support

Buhmann, Martin D.

doi:10.1017/s0013091500019416

Cited by 53 publications

(25 citation statements)

References 14 publications

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“…Several remedies to circumvent the problem were proposed. They are, in particular, domain decomposition approach [27], preconditioning and compactly supported RBF [6,26,29].…”

Section: Introductionmentioning

confidence: 99%

On using radial basis functions in a ?finite difference mode? with applications to elasticity problems

Tolstykh

Shirobokov

2003

Computational Mechanics

236

128

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A way of using RBF as the basis for PDE's solvers is presented, its essence being constructing approximate formulas for derivatives discretizations based on RBF interpolants with local supports similar to stencils in finite difference methods. Numerical results for different types of elasticity equations showing reasonable accuracy and good h-convergence properties of the technique are presented. In particular, examples of RBF solution in the case of non-linear Karman-Fopple equations are considered. IntroductionIn numerical methods for solid mechanics, the FEM method is universally accepted technique. It allows to exploit unstructured grids thus providing considerable flexibility. However, in the recent years considerable attention was paid to the so called meshless methods which operate with nodes rather than meshes. The motivation came mainly from the following considerations:-meshless methods do not require grid generation which can be not an easy task in the three dimensional cases; -meshless methods are more appropriate than FEM or finite difference methods are in the cases of very large mesh deformation and moving discontinuities.When classifying meshless methods for solving PDEs, at least two approaches can be distinguished:(i) Methods based on the least squares type of approximations used mainly in the framework of weak formulations, approximated functions and their approximations being distinct at nodes. (ii) Methods based on the interpolation principle requiring that interpolants are equal to interpolated function at nodes.To this one can add the so called generalized finite difference method for irregularly spaced nodes [3,15,20] which uses multidimensional Taylor expansion series.There are a lot of methods of type (i) described in the literature (the relevant survey and references can be found, for example, in [4]). As to (ii), it is represented by Radial Basis Functions (RBF) techniques . Considering the latter category, the existence and accuracy of scattered data RBF interpolants are widely discussed. A large body of publications concerning the subject is presented in [13].The use of RBF for PDEs discretizations offers some nice possibilities. First, some RBF-based discretizations have potential for providing convergence rates dependent on exact solutions smoothness only rather than on degrees of underlying polynomial approximations. In certain cases, they can be exponential.Second, good RBF performance in three-dimensional cases is theoretically expected.At present, the most popular lines of attack when constructing RBF-PDE solvers seem to be collocation and boundary elements approaches [7,8,12,31].In [28,30] convergence proofs and error estimates of the collocation procedure are presented. A profound impact on the RBF-collocation technique applications is due to papers [12,14,21,27]. In the works, much attention was given to Hardy's multiquadric [11] with varying shape parameters. Dramatic increase of accuracy was found in the case of properly defined shape-parameter distributions.The main diffi...

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“…Several remedies to circumvent the problem were proposed. They are, in particular, domain decomposition approach [27], preconditioning and compactly supported RBF [6,26,29].…”

Section: Introductionmentioning

confidence: 99%

On using radial basis functions in a ?finite difference mode? with applications to elasticity problems

Tolstykh

Shirobokov

2003

Computational Mechanics

236

128

View full text Add to dashboard Cite

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“…In BPNNs a common barrier is the training speed (i.e., the training speed increases as the number of layers, and number of neurons in a layer grow) [3]. To circumvent this problem a new paradigm of simpler neural network architectures with only one hidden layer has been penetrated to many application areas with a name of Radial Basis Function Neural Networks (RBFNs) [4,[6][7][8][9][10][11][12][13][14]. RBFNs were rst introduced by Powell [15][16][17][18] to solve the interpolation problem in a multi-dimensional space requiring as many centers as data points.…”

Section: Introductionmentioning

confidence: 99%

Radial basis function neural networks: a topical state-of-the-art survey

Dash¹,

Behera²,

Dehuri

et al. 2016

Open Computer Science

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Radial basis function networks (RBFNs) have gained widespread appeal amongst researchers and have shown good performance in a variety of application domains. They have potential for hybridization and demonstrate some interesting emergent behaviors. This paper aims to o er a compendious and sensible survey on RBF networks. The advantages they o er, such as fast training and global approximation capability with local responses, are attracting many researchers to use them in diversi ed elds. The overall algorithmic development of RBF networks by giving special focus on their learning methods, novel kernels, and ne tuning of kernel parameters have been discussed. In addition, we have considered the recent research work on optimization of multi-criterions in RBF networks and a range of indicative application areas along with some open source RBFN tools.

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“…As a solution of this problem, Wendland (see [18]) constructed functions φ d,ℓ = p d,ℓ χ [0,1] , for integers d ≥ 1 and ℓ ≥ 0, such that φ d,ℓ • ρ d has a nonnegative Fourier transform and belongs to C 2ℓ (R d ), the degree of the polynomial p d,ℓ being minimal. Other noteworthy constructions are those of Buhmann [4], who constructed "single-piece" piecewise functions of the form φ • ρ d , where φ = qχ [0,1] with q analytic on (0, 1], as well as several families constructed by Gneiting (see [8] and the references therein). Recently, Al-Rashdan and the author (see [2]) showed that the B-spline ψ k , having simple knots at {±1, ±2, .…”

Section: Introductionmentioning

confidence: 99%

“…The function η 4,3 = Dη 3 has regularity (4,3,0) and the first piece of η 4,3 equals 1 5 (−b 12 + (5 − 2b 14 )t 2 + 20t 2 log t). It follows that η 4,3 ∈ U 2 and therefore, by Theorem 7.4, η 4,3 has regularity (2, 2, 1).…”

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confidence: 99%

Compactly Supported, Piecewise Polyharmonic Radial Functions with Prescribed Regularity

Johnson

2011

Constr Approx

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Abstract. A compactly supported radially symmetric function Φ : R d → R is said to have Sobolev regularity k if there exist constants B ≥ A > 0 such that the Fourier transform of Φ satisfiesSuch functions are useful in radial basis function methods because the resulting native space will correspond to the Sobolev space W k 2 (R d ). For even dimensions d and integers k ≥ d/4, we construct piecewise polyharmonic radial functions with Sobolev regularity k. Two families are actually constructed. In the first, the functions have k nontrivial pieces while in the second, exactly one nontrivial piece. We also explain, in terms of regularity, the effect of restricting Φ to a lower dimensional space R d−2ℓ of the same parity.

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Radial functions on compact support

Cited by 53 publications

References 14 publications

On using radial basis functions in a ?finite difference mode? with applications to elasticity problems

On using radial basis functions in a ?finite difference mode? with applications to elasticity problems

Radial basis function neural networks: a topical state-of-the-art survey

Compactly Supported, Piecewise Polyharmonic Radial Functions with Prescribed Regularity

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