E. Hlawka, "Funktionen von Beschr ankter Variation in der Theorie der Gleichverteilung," Ann Mat. Pura Appl., vol. 54, pp. 325--333, 1961. Home/Search Document Not in Database Summary Related Articles Check |
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Unknown - Quasi-Monte Carlo Sampling (Correct)
....x ni = x n 1 i . In this setting D # n (x n1 , x nn ) O(log(n) n) is possible. The effect is like reducing d by one, but the practical cost is that such a sequence is not extensible to larger n. There is a connection between better discrepancy and more accurate integration. Hlawka [16] proved the Koksma Hlawka inequality I D # n (x 1 , x n )V HK (f) 1.17) The factor V HK (f) is the total variation of f in the sense of Hardy and Krause. Niederreiter [26] gives the definition. Equation (1.17) shows that a deterministic law of large numbers can be much better ....
E. Hlawka. Funktionen von beschr ankter Variation in der Theorie der Gleichverteilung. Annali di Matematica Pura ed Applicata, 54:325--333, 1961.
Obtaining ... Convergence for Lattice Quadrature Rules - Hickernell (Correct)
.... ## Hbv denotes the variation in the sense of Hardy and Krause, then the KoksmaHlawka inequality implies that e w (N, qmc, H bv ) D # ( x j N j=1 ) 1 O(N 2 # ) Convergence for Lattice Rules 3 for general quasi Monte Carlo rules, where D # ( x j N j=1 ) denotes the stardiscrepancy [Hla61,Nie92]. Generalizations involving the L p star discrepancy for 1 # p # # have been found as well [Zar68,Sob69,Hic98a] The first N points of a Halton sequence satisfies D # (hal) # c(s, hal)N 1 [log N ] s (see [Hal60,Nie92] Other low discrepancy sets and sequences have convergence rates ....
E. Hlawka, Funktionen von beschrankter Variation in der Theorie der Gleichverteilung, Ann. Mat. Pura Appl. 54 (1961), 325--333.
Extensible Lattice Sequences For Quasi-Monte Carlo.. - Hickernell, Hong.. (1999) (3 citations) (Correct)
....or measure of nonuniformity of the point set defining the quadrature rule, and V (f) is the variation or fluctuation of the integrand, f . The precise definitions of the discrepancy and the variation depend on the particular space of integrands. In the traditional Koksma Hlawka inequality (see [27] and [45, Theorem 2.11] the variation is the variation in the sense of Hardy and Krause, and the discrepancy is the L# star discrepancy: D # (P ) #F unif (x) FP (x)# # = # # # # x 1 . x s P # [0, x] N # # # # # . 3.2) Here F unif is the uniform distribution ....
E. Hlawka, Funktionen von beschrankter Variation in der Theorie der Gleichverteilung, Ann. Mat. Pura Appl., 54 (1961), pp. 325--333.
The Method of Fundamental Solutions and the Quasi-Monte.. - Chen, Golberg, Hon (1995) (Correct)
.... the pairwise relatively prime bases b 1 b 2 ; Delta Delta Delta ; b d Gamma1 , then D (x 0 ; x 1 ; xN ) d N 1 N d Gamma1 Y i=1 b i Gamma 1 2 log b i log N b i 1 2 : By a fundamental theory of numerical integration, we have the following inequality of Hlawka [12], which is often called the Koksma Hlawka inequality. Theorem 3.6. If f has bounded variation V (f) on I d = 0; 1] d in the sense of Hardy and Krause, then, for any x 1 ; Delta Delta Delta ; xN 2 [0; 1] d ; we have fi fi fi fi fi 1 N N X i=1 f(x n ) Gamma Z I d f(u)du ....
E. Hlawka, Funktionen von beschrankter Variation in der Theorie der Gleichverteilung, Ann. Mat. Pura Appl., 1961, 54, 325-333.
Digital (t,m,s)-nets, digital (T,s)-sequences, and.. - Larcher, Schmid, Wolf (Correct)
....1) s IR be of bounded variation V (f) in the sense of Hardy and Krause. Then for any point set x 1 ; xN of points in [0; 1) s with star discrepancy D N we have fi fi fi fi fi Z [0;1) s f(x) dx Gamma 1 N N X k=1 f (x k ) fi fi fi fi fi V (f) Delta D N : Proof See [Hla61]. This inequality, working for a quite general class of functions (especially for all continuous functions on [0; 1) s ) motivates the use of and the search for extremely well distributed point sets in the unit cube. In this connection we remind Theorem 3, the general lower bound of Roth for the ....
E. Hlawka. Funktionen von beschrankter Variation in der Theorie der Gleichverteilung. Ann. Mat. Pura Appl., 54:325--333, 1961.
Extensible Lattice Sequences For Quasi-Monte Carlo.. - Hickernell, HONG.. (1999) (3 citations) (Correct)
....or measure of nonuniformity of the point set defining the quadrature rule, and V (f) is the variation or fluctuation of the integrand, f . The precise definitions of the discrepancy and the variation depend on the particular space of integrands. In the traditional Koksma Hlawka inequality (see [26] and [44, Theorem 2.11] the variation is the variation in the sense of Hardy and Krause, and the discrepancy is the L1 star discrepancy: D (P ) kF unif (x) Gamma FP (x)k 1 = fl fl fl fl x 1 : x s Gamma jP [0; x]j N fl fl fl fl 1 : 3.2) Here F unif is the uniform distribution ....
E. Hlawka, Funktionen von beschrankter Variation in der Theorie der Gleichverteilung, Ann. Mat. Pura Appl., 54 (1961), pp. 325--333.
Pattern Recognition as a Deterministic Problem: An Approach .. - Cervellera, Muselli (Correct)
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E. Hlawka, "Funktionen von Beschr ankter Variation in der Theorie der Gleichverteilung," Ann Mat. Pura Appl., vol. 54, pp. 325--333, 1961.
Deterministic Design for Neural Network Learning: - An Approach Based (Correct)
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E. Hlawka, "Funktionen von Beschr ankter Variation in der Theorie der Gleichverteilung," Ann Mat. Pura Appl., vol. 54, pp. 325--333, 1961. 12
A Deterministic Learning Approach Based on Discrepancy - Cervellera, Muselli (Correct)
No context found.
E. Hlawka, "Funktionen von Beschrankter Variation in der Theorie der Gleichverteilung, " Ann Mat. Pura Appl., vol. 54, pp. 325--333, 1961.
Transformation of Uniformly Distributed Particle Ensembles - Hack (1995) (Correct)
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E.Hlawka, Funktionen von beschrankter Variation in der Theorie der Gleichverteilung, Annali di Matematica (Ser.IV) 54 (1961) 325--334.
Construction of Particlesets to Simulate Rarefied Gases - Hack (1993) (1 citation) (Correct)
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E.Hlawka, Funktionen von beschrankter Variation in der Theorie der Gleichverteilung, Annali di Matematica (Ser.IV) 54, pp. 325--334, 1961
Variance Reduction - Any (Correct)
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Hlawka, E., 1961, "Funktionen von Beschrankter Variation in der Theorie der Gleichverteilung," Annali di Matematica Pura Ed Applicata 54: 325-333.
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