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On the mean square weighted ℒ₂ discrepancy of randomized digital (t,m,s)-nets over ℤ₂

100%
Dick J., Pillichshammer F.
Acta Arithmetica
|
2005
|
tom 117
|
nr 4
371-403
2
Content available remote

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

100%
Aistleitner C., Dick J.
Acta Arithmetica
|
2015
|
tom 167
|
nr 2
143-171
EN
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. We also 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.
3
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Low-discrepancy point sets for non-uniform measures

100%
Aistleitner C., Dick J.
Acta Arithmetica
|
2014
|
tom 163
|
nr 4
345-369
EN
We prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure μ on the d-dimensional unit cube. We improve a theorem of Beck, by showing that for any d ≥ 1, N ≥ 1, and any non-negative, normalized Borel measure μ on $[0,1]^d$ there exists a point set $x_1, ..., x_N ∈ [0,1]^d$ whose star-discrepancy with respect to μ is of order $D_N*(x_1, ..., x_N; μ ) ≪ ((log N)^{(3d+1)/2})/N$. For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear discrepancy of hypergraphs. Furthermore, the theory of large deviation bounds for empirical processes indexed by sets is discussed, and we prove a numerically explicit upper bound for the inverse of the discrepancy for Vapnik-Chervonenkis classes. Finally, using a recent version of the Koksma-Hlawka inequality due to Brandolini, Colzani, Gigante and Travaglini, we show that our results imply the existence of cubature rules yielding fast convergence rates for the numerical integration of functions having discontinuities of a certain form.
4
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Optimal 𝓛₂ discrepancy bounds for higher order digital sequences over the finite field 𝔽₂

100%
Dick J., Pillichshammer F.
Acta Arithmetica
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2014
|
tom 162
|
nr 1
65-99
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