Standard Information for Operators

Novak, Erich; Woźniakowski, Henryk

doi:10.4171/116

Cited by 296 publications

(643 citation statements)

References 0 publications

Supporting

Mentioning

625

Contrasting

Unclassified

Order By: Relevance

“…First, we remark that we use the prefix EC (exponential convergence) in (a)-(c) in order to strictly distinguish between tractability in terms of s and log ε −1 and the classical setting where one considers s and ε −1 (cf. [20][21][22]). The term EC was introduced by Woźniakowski and is motivated by the fact that EC-polynomial tractability (and therefore also EC-strong polynomial tractability) implies uniform exponential convergence (see [6,14]).…”

Section: Introductionmentioning

confidence: 97%

“…Similar notions of tractability have been introduced by Papageorgiou and Petras [24] which they call ''polylog-tractability'' and ''ln κ -weak tractability''. Tractability in the context of exponential error convergence is also addressed in the third volume of the books of Novak and Woźniakowski [22]. Roughly speaking, we mean by EC-tractability that n(ε, s) lacks a certain disadvantageous dependence on s but also on log ε −1 .…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Integration in Hermite spaces of analytic functions

Irrgeher

Kritzer

Leobacher

et al. 2015

Journal of Complexity

View full text Add to dashboard Cite

a b s t r a c tWe study integration in a class of Hilbert spaces of analytic functions defined on the R s . The functions are characterized by the property that their Hermite coefficients decay exponentially fast. We use Gauss-Hermite integration rules and show that the errors of our algorithms decay exponentially fast. Furthermore, we study tractability in terms of s and log ε −1 and give necessary and sufficient conditions under which we achieve exponential convergence with EC-weak, EC-polynomial, and EC-strong polynomial tractability.

show abstract

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

Integration in Hermite spaces of analytic functions

Irrgeher

Kritzer

Leobacher

et al. 2015

Journal of Complexity

View full text Add to dashboard Cite

show abstract

“…This may increase the belief in the conjecture e n ≍ a n , see e.g. [8,Open Problem 140] Our improved result reads as follows.…”

mentioning

confidence: 88%

On the worst-case error of least squares algorithms for L2-approximation with high probability

Ullrich

2020

Journal of Complexity

View full text Add to dashboard Cite

It was recently shown in [4] that, for L 2 -approximation of functions from a Hilbert space, function values are almost as powerful as arbitrary linear information, if the approximation numbers are square-summable. That is, we showed thatwhere e n are the sampling numbers and a k are the approximation numbers. In particular, if (a k ) ∈ ℓ 2 , then e n and a n are of the same polynomial order. For this, we presented an explicit (weighted least squares) algorithm based on i.i.d. random points and proved that this works with positive probability. This implies the existence of a good deterministic sampling algorithm.Here, we present a modification of the proof in [4] that shows that the same algorithm works with probability at least 1 − n −c for all c > 0.Let H be a Hilbert space of real-or complex-valued functions on a set D such that point evaluationis a continuous functional for all x ∈ D, which are usually called reproducing kernel Hilbert spaces. We consider numerical approximation of functions from such spaces, using only function values. We measure the error in the space L 2 = L 2 (D, A, µ) of square-integrable functions with respect to an arbitrary measure µ such that H is embedded into L 2 . This means that H consists of square-integrable functions such that two functions that are equal µ-almost everywhere are also equal point-wise.

show abstract

“…A comprehensive theory of general tractability was developed by Gnewuch and Woźniakowski [24,25]. The books of Novak and Woźniakowski [39][40][41] summarize the current state of the art in tractability theory.…”

Section: Applications and Constructionsmentioning

confidence: 99%