Sampling and Stability

Rieger, Christian; Schaback, Robert; Zwicknagl, Barbara

doi:10.1007/978-3-642-11620-9_23

Cited by 17 publications

(18 citation statements)

References 28 publications

(18 reference statements)

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“…. , x N } that are Π d s -unisolvent and all vectors a ∈ R N that satisfy zero moment conditions on X in the sense of (8), that is λ a,X p = 0 for each p ∈ Π d s . We can define an inner product on these functionals by…”

Section: Introductionmentioning

confidence: 99%

Error bounds for kernel-based numerical differentiation

Davydov

Schaback

2015

Numer. Math.

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The literature on meshless methods observed that kernel-based numerical differentiation formulae are robust and provide high accuracy at low cost. This paper analyzes the error of such formulas, using the new technique of growth functions. It allows to bypass certain technical assumptions that were needed to prove the standard error bounds on interpolants and their derivatives. Since differentiation formulas based on polynomials also have error bounds in terms of growth functions, we have a convenient way to compare kernel-based and polynomial-based formulas. It follows that kernel-based formulas are comparable in accuracy to the best possible polynomial-based formulas. A variety of examples is provided.

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Section: Introductionmentioning

confidence: 99%

Error bounds for kernel-based numerical differentiation

Davydov

Schaback

2015

Numer. Math.

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“…A few articles deal with the case Ω unbounded (Madych and Potter [12], Madych [11] and Arcangéli et al [3]). For a survey on sampling inequalities and their applications, the reader is referred to Rieger et al [17]. The purpose of this paper is to extend the results of [2] and [3], for both cases Ω bounded and Ω = R n , in two directions: to allow fractional order semi-norms on the left-hand side of the sampling inequality (1.1), and to admit derivative pointwise values on the second term of the right-hand side of (1.1), so increasing the order of the approximation parameter d.…”

mentioning

confidence: 99%

Extension of sampling inequalities to Sobolev semi-norms of fractional order and derivative data

Arcangéli¹,

Silanes

Torrens

2011

Numer. Math.

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This paper, devoted to sampling inequalities, provides some extensions of previous results by Arcangéli et al. (Numer Math 107(2):181-211, 2007; J Approx Theory 161:198-212, 2009). Given a function u in a suitable Sobolev space defined on a domain Ω in R n , sampling inequalities typically yield bounds of integer order Sobolev semi-norms of u in terms of a higher order Sobolev semi-norm of u, the fill distance d between Ω and a discrete set A ⊂ Ω, and the values of u on A. The extensions established in this paper allow us to bound fractional order semi-norms and to incorporate, if available, values of partial derivatives on the discrete set. Both the cases of a bounded domain Ω and Ω = R n are considered.

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“…We now use the much more recent framework of sampling inequalities in reproducing kernel Hilbert spaces (see, e.g. [31]) to obtain the same error bounds as already reported in [42]. The following discussion closely follows [31,34] and also draws on some ingredients from [42].…”

Section: Error Bounds Via Sampling Inequalitiesmentioning

confidence: 93%

An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels

2014

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Kernel-based approximation methods-often in the form of radial basis functions-have been used for many years now and usually involve setting up a kernel matrix which may be ill-conditioned when the shape parameter of the kernel takes on extreme values, i.e., makes the kernel "flat". In this paper we present an algorithm we refer to as the Hilbert-Schmidt SVD and use it to emphasize two important points which-while not entirely new-present a paradigm shift under way in the practical application of kernel-based approximation methods: (i) it is not necessary to form the kernel matrix (in fact, it might even be a bad idea to do so), and (ii) it is not necessary to know the kernel in closed form. While the Hilbert-Schmidt SVD and its two implications apply to general positive definite kernels, we introduce in this paper a class of so-called iterated Brownian bridge kernels which allow us to keep the discussion as simple and accessible as possible.

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Sampling and Stability

Cited by 17 publications

References 28 publications

Error bounds for kernel-based numerical differentiation

Error bounds for kernel-based numerical differentiation

Extension of sampling inequalities to Sobolev semi-norms of fractional order and derivative data

An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels

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