Radial basis functions under tension

Bouhamidi, A.; Méhauté, Alain Le

doi:10.1016/j.jat.2004.03.005

Cited by 13 publications

(10 citation statements)

References 16 publications

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“…According to the Sobolev inequality [1] and [21,Theorem 10.40], H P (R) has the Sdense property which implies that N 1 G (R) ≡ H P (R). Theorem 4.4 and [21, Theorem 13.2] provide us with the same optimality property as stated in [4,16].…”

Section: Example 52 (Tension Splines)mentioning

confidence: 96%

“…So G is a conditionally positive definite function of order 1. This yields the same interpolant as the tension spline interpolant [4,16]. According to the Sobolev inequality [1] and [21,Theorem 10.40], H P (R) has the Sdense property which implies that N 1 G (R) ≡ H P (R).…”

Section: Example 52 (Tension Splines)mentioning

confidence: 96%

“…A large and increasing number of recent books and research papers apply radial basis functions or other kernel-based approximation methods to such fields as scattered data approximation, statistical or machine learning and the numerical solution of partial differential equations, e.g., [2,3,4,5,7,10,14,15,20,21]. Generally speaking, the fundamental underlying practical problem common to many of these applications can be represented in the following way.…”

Section: Introductionmentioning

confidence: 99%

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Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators

Fasshauer

2011

Numer. Math.

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In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator P consisting of finitely or countably many distributional operators P n , which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function G with respect to L := P * T P now becomes a conditionally positive function. In order to support this claim we ensure that the distributional adjoint operator P * of P is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function G can be embedded into or even be equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant s f,X to data values sampled from an unknown generalized Sobolev function f at data sites located in some set X ⊂ R d . We provide several examples, such as Matérn kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are equivalent to a generalized Sobolev space. These examples further illustrate how we can rescale the Sobolev spaces by the vector distributional operator P. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the "best" kernel function for kernel-based approximation methods.

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Section: Example 52 (Tension Splines)mentioning

confidence: 96%

Section: Example 52 (Tension Splines)mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators

Fasshauer

2011

Numer. Math.

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“…The radial basis functions under tension (RBFT) was introduced in [5]. The RBFT depend on a positive parameter and provide a convenient way of controlling the behavior of the interpolating surface.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, this fundamental solution does not have a similar explicit expression for all d ≥ 1. The crucial question now is how to construct a (pseudo-)differential operator with a fundamental solution which generates a tempered distribution that has a simple and similar explicit expression for all dimension d ≥ 1 and that allows the construction of the radial basis functions under tension as presented in [5]. The goal is the construction of a radial basis function depending on a tension parameter such that, as in the univariate case, in the limit the radial basis functions under tension reduce to the pseudo-linear splines as the tension parameter becomes large and reduce to the pseudo-cubic splines as the tension parameter becomes small.…”

Section: Introductionmentioning

confidence: 99%

Pseudo-differential operator associated to the radial basis functions under tension.

Bouhamidi

2007

ESAIM: Proc.

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Radial basis functions under tension (RBFT) depend on a positive parameter, incorporate the concept of spline with tension and provide a convenient way for the control of the behavior of the interpolating surface. The RBFT involve a function which is not complicated than exponential and may be easily coded. In this paper, we show that the RBFT, as like thin plate spline, may be associated to a differential operator in a Beppo-Levi space type. Both smoothing and interpolating problems by RBFT are studied. Résumé. Les fonctions splines radiales sous tension dépendent d'un paramètre positif. Ces fonctions permettent d'incorporer un concept de tension pour toute dimension de l'espace. On montre dans ce papier que ces fonctions sont associéesà un opérateur pseudo-différentiel dans un espace de type Beppo-Levi et onétudie le problème d'interpolation et de lissage. Des examples numériques sont donnés pour illustrer ce type d'approximation.

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