Quasi-polynomial tractability

Gnewuch, Michael; Woniakowski, Henryk

doi:10.1016/j.jco.2010.07.001

Cited by 52 publications

(42 citation statements)

References 12 publications

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“…Obviously, the identity I d is a compact tensor product operator (considered as a mapping into the tensor product space L 2 (T d )). Since the approximation numbers decay polynomially in these four univariate situations and 1 = a 1 > a 2 , we obtain the following conclusion from the general Theorem 3.3 of [8]: For any s > 0, all four problems…”

Section: Tractability Results For Hmentioning

confidence: 60%

“…We will obtain "quasi-polynomial tractability" of the respective approximation problems. This notion has been recently introduced in [8] and is a stronger notion than "weak tractability" .…”

Section: Quasi-polynomial Tractabilitymentioning

confidence: 99%

“…In both cases we have I d = 1 for all s > 0 and d ∈ N. In other words, the normalized error criterion is satisfied. In this context, a linear algorithm that uses arbitrary information is of the form We need a further notion of tractability, namely quasi-polynomial tractability, see for instance [8] . In fact, the approximation problem is called quasi-polynomially tractable if there are positive numbers t and C t such that n(ε, d) ≤ C t exp(t ln(ε −1 )(1 + ln(d))) .…”

Section: General Notions Of Tractabilitymentioning

confidence: 99%

See 2 more Smart Citations

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

Kühn

Sickel

Ullrich³

2015

Constr Approx

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We investigate the approximation of d-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness s > 0 on the d-dimensional torus, where the approximation error is measured in the L 2 −norm. In other words, we study the approximation numbers of the Sobolev embeddings H, with particular emphasis on the dependence on the dimension d. For any fixed smoothness s > 0, we find the exact asymptotic behavior of the constants as d → ∞. We observe super-exponential decay of the constants in d, if n, the number of linear samples of f , is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by low rank approximations. We present some surprising results for the socalled "preasymptotic" decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.

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Section: Tractability Results For Hmentioning

confidence: 60%

Section: Quasi-polynomial Tractabilitymentioning

confidence: 99%

Section: General Notions Of Tractabilitymentioning

confidence: 99%

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

Kühn

Sickel

Ullrich³

2015

Constr Approx

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“…As in [2], we say that INT = {INT d } is T -tractable iff there are nonnegative numbers C and t such that…”

Section: Generalized Tractability and Uniform Weak Tractabilitymentioning

confidence: 99%

Uniform Weak Tractability of Weighted Integration

Siedlecki

2016

Springer Proceedings in Mathematics &Amp; Statistics

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We study a relatively new notion of tractability called "uniform weak tractability" that was recently introduced in (Siedlecki, J. Complex. 29:438-453, 2013 [5]). This notion holds for a multivariable problem iff the information complexity n(ε, d) of its d-variate component to be solved to within ε is not an exponential function of any positive power of ε −1 and/or d. We are interested in necessary and sufficient conditions on uniform weak tractability for weighted integration. Weights are used to control the "role" or "importance" of successive variables and groups of variables. We consider here product weights. We present necessary and sufficient conditions on product weights for uniform weak tractability for two Sobolev spaces of functions defined over the whole Euclidean space with arbitrary smoothness, and of functions defined over the unit cube with smoothness 1. We also briefly consider (s, t)-weak tractability introduced in (Siedlecki and Weimar, J. Approx. Theory 200:227-258, 2015 [6]), and show that as long as t > 1 then this notion holds for weighted integration defined over quite general tensor product Hilbert spaces with arbitrary bounded product weights.

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“…However, it may happen that decay λ = 0. It is known, see [1], that S is QPT for Λ all iff λ 2 < λ 1 and decay λ > 0.…”

Section: Introductionmentioning

confidence: 99%

Tractability of Multivariate Problems for Standard and Linear Information in the Worst Case Setting: Part II

Woźniakowski¹,

Novak²

2018

Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

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Dedicated to Ian H. Sloan on the occassion of his 80th birthday. AbstractWe study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well defined. We prove QPT for algorithms that use only function values under the three assumptions:1. the minimal errors for the univariate case decay polynomially fast to zero, 2. the largest singular value for the univariate case is simple and 3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.

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Quasi-polynomial tractability

Cited by 52 publications

References 12 publications

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

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

Uniform Weak Tractability of Weighted Integration

Tractability of Multivariate Problems for Standard and Linear Information in the Worst Case Setting: Part II

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