Introduction to Quasi-Monte Carlo Integration and Applications

Leobacher, Gunther; Pillichshammer, Friedrich

doi:10.1007/978-3-319-03425-6

Cited by 149 publications

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“…. , b s−1 are mutually co-prime, then the discrepancy of the Hammersley point set satisfies [18,54,68]). …”

Section: Example: Hammersley Point Setsmentioning

confidence: 95%

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Proof techniques in quasi-Monte Carlo theory

Dick

Hinrichs

Pillichshammer

2015

Journal of Complexity

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“…. , b s−1 are mutually co-prime, then the discrepancy of the Hammersley point set satisfies [18,54,68]). …”

Section: Example: Hammersley Point Setsmentioning

confidence: 95%

“…This sequence is the prototype of many other digital constructions of point sets and sequences. It is well-known that the discrepancy of van der Corput sequences satisfies [18,45,54,68]). …”

Section: Example: Van Der Corput Sequencesmentioning

confidence: 96%

Proof techniques in quasi-Monte Carlo theory

Dick

Hinrichs

Pillichshammer

2015

Journal of Complexity

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“…But it was later proposed in 2005 [45], in part motivated by the lower-dimensional exact Bures results reported in [34], that the value might be 1680σ Ag π 8 ≈ 0.07334, where σ Ag = √ 2 − 1 ≈ 0.414214 is the "silver mean". Both of these studies [44,45] were conducted using quasi-Monte Carlo procedures, before the developmentof the indicated Osipov-Sommers-Życzkowski methodology (2) We now importantly examine the question of whether Bures two-qubit and two-rebit separability probability estimation can be accelerated-with superior convergence propertiesby, rather than using, as typically done, independently-generated normal variates for the Ginibre ensembles at each iteration, making use of normal variates jointly-generated by employing low-discrepancy (quasi-Monte Carlo) sequences [46]. In particular, we have…”

Section: B Prior Estimations Of Bures Separability Probabilitiesmentioning

confidence: 99%

Numerical and exact analyses of Bures and Hilbert–Schmidt separability and PPT probabilities

Slater

2019

Quantum Inf Process

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“…and can be used to obtain bounds on e(H Kα,γ , P ) for any α > 1/2 (see [15,Theorem 5.5] or [14,Lemma 4.20] for the case where γ 1 = . .…”

Section: This Yieldsmentioning

confidence: 99%

Numerical integration in log-Korobov and log-cosine spaces

et al. 2015

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Quasi-Monte Carlo (QMC) rules are equal weight quadrature rules for approximating integrals over the s-dimensional unit cube [0, 1] s . One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on [0, 1] s with square integrable partial mixed derivatives of order α. Using Parseval's identity, this smoothness can be defined for all real numbers α > 1/2. In this setting, the condition α > 1/2 is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined.This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with α arbitrarily close to 1/2. To obtain such estimates we introduce a log-scale for functions with smoothness close to 1/2, which we call log-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order O(N −1/2 (log N ) −µ(1−λ)/2 ) for any 1/µ < λ ≤ 1, where µ > 1 is a certain power in the log-scale.A second result is concerned with tractability of numerical integration for weighted Korobov spaces with product weights (γ j ) j∈N . Previous results have shown that if ∞ j=1 γ τ j < ∞ for some 1/(2α) < τ ≤ 1 one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where τ is close to 1/(2α), namely we show dimension independent error bounds under the condition that ∞ j=1 γ j max{1, log γ −1 j } µ(1−λ) < ∞ for some 1/µ < λ ≤ 1. The essential tool in our analysis is a log-scale Jensen's inequality.The results described above also apply to integration in log-cosine spaces using tent-transformed lattice rules.

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Introduction to Quasi-Monte Carlo Integration and Applications

Cited by 149 publications

References 0 publications

Proof techniques in quasi-Monte Carlo theory

Proof techniques in quasi-Monte Carlo theory

Numerical and exact analyses of Bures and Hilbert–Schmidt separability and PPT probabilities

Numerical integration in log-Korobov and log-cosine spaces

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