N-Widths and ε-Dimensions for High-Dimensional Approximations

Dũng, Đinh; Ullrich, Tino

doi:10.1007/s10208-013-9149-9

Cited by 37 publications

(66 citation statements)

References 28 publications

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“…• Closest to us in aims and methods is the recent paper [7]. There, in Theorem 3.13, the authors obtained for s > 0 and any n ≥ 2 d the inequality • Super-exponential decay of the constants C s (d) in d has been observed before.…”

Section: Various Comments On the Literaturementioning

confidence: 90%

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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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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: Various Comments On the Literaturementioning

confidence: 90%

“…These studies have been continued in Griebel, Knapek [10,11], Bungartz, Griebel [4], Griebel [9], Schwab, Süli, and Todor [23], and Dũng, Ullrich [7]. Let us comment on the non-periodic situation first.…”

Section: Consequently Inequality (431) Implies For Anymentioning

confidence: 99%

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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“…[43,48]. The properties of such approximations of functions in Sobolev spaces on the n-dimensional torus T n have been studied by several authors [7,9,15,31,32,34,36,41,44,45,48]. In particular, spaces of generalized mixed Sobolev smoothness H t,r mix (T n ) := f :…”

Section: Michael Griebelmentioning

confidence: 99%

Fast Discrete Fourier Transform on Generalized Sparse Grids

Griebel

Hamaekers

2014

Lecture Notes in Computational Science and Engineering

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In this paper, we present an algorithm for trigonometric interpolation of multivariate functions on generalized sparse grids and study its application for the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. In particular, we derive estimates for the error and the cost. We construct interpolants with a computational cost complexity which is substantially lower than for the standard full grid case. The associated generalized sparse grid interpolants have the same approximation order as the standard full grid interpolants, provided that certain additional regularity assumptions on the considered functions are fulfilled. Numerical results validate our theoretical findings.

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“…in the smaller spaces H s mix (T d ) of dominating mixed smoothness, see also the paper by Dũng and Ullrich [4] in this context (but these authors used different norms). The error was measured in L 2 (T d ), with particular emphasis on the constants and their dependence on the dimension d. In [8,9] the exact decay rate of the constants as d → ∞ was found, which turned out to be polynomial in d for the isotropic spaces, and superexponential in d for the mixed spaces.…”

Section: Introductionmentioning

confidence: 99%