Tensor power sequences and the approximation of tensor product operators

Krieg, David

doi:10.1016/j.jco.2017.09.002

“…5]. In addition, combining the above result with recent preasymptotic estimates for the (σ j ) j , see [14,[17][18][19], we are able to obtain reasonable bounds for g n also in the case of small n. See Sect. 7 for further comments and references in this direction.…”

Section: Introductionsupporting

confidence: 60%

“…Proof Let m ≥ 2. Similarly, as in [13,15] and [24], we use the density function (14) in order to consider the embedding Id :…”

Section: Theorem 61 Let H (K ) Be a Separable Reproducing Kernel Hilbert Space On A Set D ⊂ R D With A Positive Semidefinite Kernel Kmentioning

confidence: 99%

A New Upper Bound for Sampling Numbers

Nagel

¹

,

Schäfer

²

,

Ullrich

³

2021

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We provide a new upper bound for sampling numbers $$(g_n)_{n\in \mathbb {N}}$$ ( g n ) n ∈ N associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants $$C,c>0$$ C , c > 0 (which are specified in the paper) such that $$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$ g n 2 ≤ C log ( n ) n ∑ k ≥ ⌊ c n ⌋ σ k 2 , n ≥ 2 , where $$(\sigma _k)_{k\in \mathbb {N}}$$ ( σ k ) k ∈ N is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding $$\mathrm {Id}:H(K) \rightarrow L_2(D,\varrho _D)$$ Id : H ( K ) → L 2 ( D , ϱ D ) . The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of $$H^s_{\text {mix}}(\mathbb {T}^d)$$ H mix s ( T d ) in $$L_2(\mathbb {T}^d)$$ L 2 ( T d ) with $$s>1/2$$ s > 1 / 2 . We obtain the asymptotic bound $$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$ g n ≤ C s , d n - s log ( n ) ( d - 1 ) s + 1 / 2 , which improves on very recent results by shortening the gap between upper and lower bound to $$\sqrt{\log (n)}$$ log ( n ) . The result implies that for dimensions $$d>2$$ d > 2 any sparse grid sampling recovery method does not perform asymptotically optimal.

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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: 86%

Algorithms and complexity for functions on general domains

Novak

¹

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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“…An overview about the behaviour of approximation numbers of embeddings of Sobolev or Besov spaces into L p -spaces can be found in the monographs by Edmunds, Triebel [13] (isotropic case), Temlyakov [35], [36], Triebel [39] and Dũng, Temlyakov, Ullrich [12]. Up to now only in a few situations one knows estimates for approximation numbers of embeddings of Sobolev or Besov spaces into L p -spaces with explicit dependence on d, we refer to the papers by Krieg [16], Kühn, Sickel, Ullrich [19], [20], Kühn, Mayer, Ullrich [18], the present authors [8], and Mieth [23]. Recently Bachmayr et al [5] have investigated the approximation of tensor products of functions belonging to W t ∞ (0, 1) in the L ∞ -norm, where t is a given natural number, see also Novak, Rudolf [24] and Krieg, Rudolf [17].…”

Section: On the Approximation Of Tensor Products Of Functionsmentioning

confidence: 99%

“…From the practical point of view this is the more important range. First results for the so-called preasymptotic range have been obtained in [6], [32], [19], [20], [18] and [16]. However, up to now related results for a n (I d : H t mix (T d ) → L ∞ (T d )) are not known.…”

Section: On the Approximation Of Tensor Products Of Functionsmentioning

confidence: 99%

On optimal approximation in periodic Besov spaces

Cobos

¹

,

Kühn

²

,

Sickel

³

2019

Journal of Mathematical Analysis and Applications

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We work with spaces of periodic functions on the d-dimensional torus. We show that estimates for L ∞ -approximation of Sobolev functions remain valid when we replace L ∞ by the isotropic periodic Besov space B 0 ∞,1 or the periodic Besov space with dominating mixed smoothness S 0 ∞,1 B. For t > 1/2, we also prove estimates for L 2 -approximation of functions in the Besov space of dominating mixed smoothness S t 1,∞ B, describing exactly the dependence of the involved constants on the dimension d and the smoothness t. d j=1 |k j | r s/r , the space F d (w) coincides with the isotropic Sobolev space H s,r (T d ). The superscript r indicates the norm we are considering in the Sobolev space. When w(k) = d j=1 1 + |k j | r s/r , we obtain the Sobolev space of dominating mixed smoothness H s,r mix (T d ) (see [8] and [20]).

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Tensor power sequences and the approximation of tensor product operators

Cited by 17 publications

References 13 publications

A New Upper Bound for Sampling Numbers

A New Upper Bound for Sampling Numbers

Algorithms and complexity for functions on general domains

On optimal approximation in periodic Besov spaces

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