(<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mo>ln</mml:mo></mml:mrow><mml:mrow><mml:mi>κ</mml:mi></mml:mrow></mml:msup></mml:math>)-weak tractability of linear problems

Papageorgiou, Anargyros; Petras, Iasonas; Woźniakowski, Henryk

doi:10.1016/j.jco.2016.10.008

Cited by 9 publications

(9 citation statements)

References 8 publications

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“…Finding necessary and sufficient conditions on the λ j for the different kinds of exponential tractability turns out to be a technically difficult question. Necessary and sufficient conditions have been considered in the paper [5]. Here we re-prove the results in [5], using a completely different technique, namely using criteria that have been shown very recently in the paper [1].…”

Section: Introductionmentioning

confidence: 60%

“…Here, we deal with the case of tensor product problems for which the singular values of a dvariate problem are given as products of the singular values of univariate problems. Exponential tractability for tensor product problems has been studied in [5]. In this paper we re-prove the results of [5] by a different argument via the criteria presented in [1].…”

Section: Introductionmentioning

confidence: 67%

“…Exponential tractability for tensor product problems has been studied in [5]. In this paper we re-prove the results of [5] by a different argument via the criteria presented in [1].…”

Section: Introductionmentioning

confidence: 67%

“…Necessity of (8): The necessity of (8) for the corresponding subcases follows from Items L1 and L2 of Lemma 1 in [5]. We remark that these are only based on general definitions, and do not require the technical results used in the proof of the main theorem of [5].…”

Section: Necessary Conditionsmentioning

confidence: 99%

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Exponential Tractability of Linear Tensor Product Problems

2020

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In this article we consider the approximation of compact linear operators defined over tensor product Hilbert spaces. Necessary and sufficient conditions on the singular values of the problem under which we can or cannot achieve different notions of exponential tractability are given in [5]. In this paper, we use the new equivalency conditions shown in [1] to obtain these results in an alternative way. As opposed to the algebraic setting, quasi-polynomial tractability is not possible for non-trivial cases in the exponential setting.

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Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 67%

“…Exponential tractability for tensor product problems has been studied in [5]. In this paper we re-prove the results of [5] by a different argument via the criteria presented in [1].…”

Section: Introductionmentioning

confidence: 67%

Section: Necessary Conditionsmentioning

confidence: 99%

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Exponential Tractability of Linear Tensor Product Problems

2020

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“…However, there is also a recent stream of work on exponential tractability, and this is what we are going to study here. For results on exponential tractability for general linear problems (without necessarily assuming tensor product structure), we refer the reader to the recent paper [12], for exponential tractability for linear problems on un-weighted tensor product spaces, we refer the reader to [4,21,22]. Further results on exponential tractability can, e.g., be found in the papers [1,2,6,7,8,10,11,13,14,15,16,23,27,28,31,33].…”

Section: Introductionmentioning

confidence: 99%

Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals

Kritzer¹,

Pillichshammer²,

Woźniakowski³

2020

Preprint

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We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals which is needed to obtain an ε-approximation for the d-variate problem. It is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function which depends polynomially on d and logarithmically on ε −1 . The corresponding un-weighted problem was studied in [4] with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on ε −1 , we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential (s, t)-weak tractability with max(s, t) ≥ 1. For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential (s, t)-weak tractability with max(s, t) < 1 is left for future study. The paper uses some general results obtained in [4] and [12].

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Approximation numbers of Sobolev and Gevrey type embeddings on the sphere and on the ball—Preasymptotics, asymptotics, and tractability

Chen

Wang

2019

Journal of Complexity

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In this paper, we investigate optimal linear approximations (napproximation numbers ) of the embeddings from the Sobolev spaces H r (r > 0) for various equivalent norms and the Gevrey type spaces G α,β (α, β > 0) on the sphere S d and on the ball B d , where the approximation error is measured in the L 2 -norm. We obtain preasymptotics, asymptotics, and strong equivalences of the above approximation numbers as a function in n and the dimension d. We emphasis that all equivalence constants in the above preasymptotics and asymptotics are independent of the dimension d and n. As a consequence we obtain that for the absolute error criterion the approximation problems I d : H r → L 2 are weakly tractable if and only if r > 1, not uniformly weakly tractable, and do not suffer from the curse of dimensionality. We also prove that for any α, β > 0, the approximation problems I d : G α,β → L 2 are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if α ≥ 1. r/d exist, having the same value for various norms. We also prove that for 0 < α < 1,, lim n→∞ e βγn α/d a n (I d : G α,β (S d ) → L 2 (S d )) = 1.

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(s,lnκ)-weak tractability of linear problems

Cited by 9 publications

References 8 publications

Exponential Tractability of Linear Tensor Product Problems

Exponential Tractability of Linear Tensor Product Problems

Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals

Approximation numbers of Sobolev and Gevrey type embeddings on the sphere and on the ball—Preasymptotics, asymptotics, and tractability

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