Approximation of Mixed Order Sobolev Functions on the d-Torus: Asymptotics, Preasymptotics, and d-Dependence

Kühn, Thomas; Sickel, Winfried; Ullrich, Tino

doi:10.1007/s00365-015-9299-x

Cited by 68 publications

(88 citation statements)

References 23 publications

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“…s n −s (log n) s(d−1) . Theorem 2 is due to Theorem 4.3 in [KSU15]. There, Kühn, Sickel and Ullrich prove this asymptotic equality in an interesting special case: τ is the sequence of approximation numbers for the L 2 -embedding of the tensor power space H s mix T d on the d-torus [0, 2π] d , equipped with a tensor product norm.…”

Section: Introduction and Resultsmentioning

confidence: 95%

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Tensor power sequences and the approximation of tensor product operators

Krieg

2018

Journal of Complexity

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The approximation numbers of the L 2 -embedding of mixed order Sobolev functions on the d-torus are well studied. They are given as the nonincreasing rearrangement of the dth tensor power of the approximation number sequence in the univariate case. I present results on the asymptotic and preasymptotic behavior for tensor powers of arbitrary sequences of polynomial decay. This can be used to study the approximation numbers of many other tensor product operators, like the embedding of mixed order Sobolev functions on the d-cube into L 2 [0, 1] d or the embedding of mixed order Jacobi functions on the d-cube into L 2 [0, 1] d , w d with Jacobi weight w d .

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Section: Introduction and Resultsmentioning

confidence: 95%

“…The statement can be deduced from this special case with the help of their Lemma 4.14. However, we prefer to give a direct proof in Section 2 by generalizing the proof of Theorem 4.3 in [KSU15].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Tensor power sequences and the approximation of tensor product operators

Krieg

2018

Journal of Complexity

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“…like the cube, see [6,20,23,24]. Quite remarkable are recent results of Mieth [26], to be discussed later, since they hold for general domains D d .…”

Section: Introductionmentioning

confidence: 85%

Algorithms and complexity for functions on general domains

Novak

2020

Journal of Complexity

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Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the approximation or integration of functions defined on D d ⊂ R d and only assume that D d is a bounded Lipschitz domain. Some results are even more general. We study three different concepts to measure the complexity: order of convergence, asymptotic constant, and explicit uniform bounds, i.e., bounds that hold for all n (number of pieces of information) and all (normalized) domains.It is known for many problems that the order of convergence of optimal algorithms does not depend on the domain D d ⊂ R d . We present examples for which the following statements are true:1. Also the asymptotic constant does not depend on the shape of D d or the imposed boundary values, it only depends on the volume of the domain.2. There are explicit and uniform lower (or upper, respectively) bounds for the error that are only slightly smaller (or larger, respectively) than the asymptotic error bound. *Much is known about the order of convergence of the numbers e n , in particular for simple domains D d , such as the cube or the torus. General bounded Lipschitz domains are studied in [7,10,11,12,26,27,29,38,39,40,42,43]. In many cases, the optimal order of the e n is of the form e n ≍ n −α (log n) β , where α and β do not depend on the domain D d . It is interesting to know also the exact asymptotic constant C := lim n→∞ e n n α (log n) −β , if it exists. The value of C is known only in rare cases (unless d = 1, we do not discuss the univariate case in detail), and usually only for very special domains,

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“…Further developments in the periodic setting of Sobolev spaces with fractional smoothness and with mixed order smoothness were made in [5,14,16]. In the non-periodic case even less is known.…”

Section: Introductionmentioning

confidence: 99%