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Woźniakowski, Henryk

doi:10.4171/026

Cited by 340 publications

(404 citation statements)

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“…More recent results may be found in Temlyakov's monograph [29]. The role of the spaces H s mix (T d ) in IBC is summarized in the recent series of books by Novak and Woźniakowski [17,19,20]. Observe that in IBC the spaces are sometimes called Korobov spaces, see, e.g., [17, pp.…”

Section: Isotropic Sobolev Spaces On the D-torusmentioning

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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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: Isotropic Sobolev Spaces On the D-torusmentioning

confidence: 99%

“…Various concepts of tractability are discussed in the recent monographs by Novak and Woźniakowski [17,19,20] . We will obtain "quasi-polynomial tractability" of the respective approximation problems.…”

Section: Quasi-polynomial Tractabilitymentioning

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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“…Corollary 3 implies that the problem of integrating d-dimensional functions whose HK-variation is uniformly bounded with respect to any normalized measure is polynomially tractable. For more information on tractability see the three volumes on Tractability of Multivariate Problems by Novak and Woźniakowski [31,32,33]. In the case of the function class being restricted to indicator functions of sets A ∈ A * (that is, in the case of the left-hand side of (13) being the star-discrepancy with respect to µ) Corollary 3 was proved in [18] (without an effective value for the constant in (12)).…”

Section: Corollary 2 For Any Normalized Borel Measurementioning

confidence: 99%

Functions of bounded variation, signed measures, and a general Koksma–Hlawka inequality

Aistleitner

Dick

2015

Acta Arith.

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Abstract. In this paper we prove a correspondence principle between multivariate functions of bounded variation in the sense of Hardy and Krause and signed measures of finite total variation, which allows us to obtain a simple proof of a generalized Koksma-Hlawka inequality for non-uniform measures. Applications of this inequality to importance sampling in Quasi-Monte Carlo integration and tractability theory are given. Furthermore, we discuss the problem of transforming a low-discrepancy sequence with respect to the uniform measure into a sequence with low discrepancy with respect to a general measure µ, and show the limitations of a method suggested by Chelson.

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“…Let us first recall some related notions, see [19,34]. LetF and G be normed spaces of functions defined on the set D d ⊂ R d .…”

Section: Introductionmentioning

confidence: 99%

Gelfand numbers of embeddings of mixed Besov spaces

Nguyen

2017

Journal of Complexity

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Gelfand numbers represent a measure for the information complexity which is given by the number of information needed to approximate functions in a subset of a normed space with an error less than ε. More precisely, Gelfand numbers coincide up to the factor 2 with the minimal error e wor (n, Λ all ) which describes the error of the optimal (non-linear) algorithm that is based on n arbitrary linear functionals. This explains the crucial role of Gelfand numbers in the study of approximation problems. Let S t p1,p1 B((0, 1) d ) be the Besov spaces with dominating mixed smoothness on (0, 1)d . In this paper we consider the problem App : 0, 1) d ) and investigate the asymptotic behaviour of Gelfand numbers of this embedding. We shall give the correct order of convergence of Gelfand numbers in almost all cases. In addition we shall compare these results with the known behaviour of approximation numbers which coincide with e wor−lin (n, Λ all ) when we only allow linear algorithms.

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Cited by 340 publications

References 0 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

Functions of bounded variation, signed measures, and a general Koksma–Hlawka inequality

Gelfand numbers of embeddings of mixed Besov spaces

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