Low-discrepancy sequences: Atanassov’s methods revisited
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- 作者：Henri Faurea ; henri.faure@univ-amu.fr" class="auth_mail" title="E-mail the corresponding author ; Christiane Lemieuxb ; clemieux@uwaterloo.ca" class="auth_mail" title="E-mail the corresponding author
- 关键词：Discrepancy bounds ; Low-discrepancy sequences ; Atanassov&rsquo ; s methods
- 刊名：Mathematics and Computers in Simulation
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：132
- 期：Complete
- 页码：236-256
- 全文大小：431 K
文摘
We revisit Atanassov’s methods for Halton sequences, 75416301513&_mathId=si1.gif&_user=111111111&_pii=S0378475416301513&_rdoc=1&_issn=03784754&md5=ded6e8dfc41c5a3cb2007a381c136ffb" title="Click to view the MathML source">(t,s)-sequences, and 75416301513&_mathId=si7.gif&_user=111111111&_pii=S0378475416301513&_rdoc=1&_issn=03784754&md5=0a90897f7d747f9599161974fbd3eba2">
75416301513-si7.gif">-sequences by providing a unifying framework enlightening the power and the efficiency of these methods for the study of low-discrepancy sequences. In this context, we obtain new improved explicit bounds for the star-discrepancy of these sequences, showing in most cases a better behavior than preceding ones in the non-asymptotic regime. Theoretical comparisons of discrepancy bounds in the non-asymptotic regime are much more difficult to achieve than in the asymptotic regime, where results exist to compare the leading constants 75416301513&_mathId=si15.gif&_user=111111111&_pii=S0378475416301513&_rdoc=1&_issn=03784754&md5=6fd84028d912d1bf8fac27b023107eb4" title="Click to view the MathML source">cs. Hence in this paper we mostly proceed via numerical comparisons to compare bounds. But in the case of 75416301513&_mathId=si1.gif&_user=111111111&_pii=S0378475416301513&_rdoc=1&_issn=03784754&md5=ded6e8dfc41c5a3cb2007a381c136ffb" title="Click to view the MathML source">(t,s)-sequences in base 2, we are able to compare two discrepancy bounds and prove that one is demonstrably better than the other for any 75416301513&_mathId=si17.gif&_user=111111111&_pii=S0378475416301513&_rdoc=1&_issn=03784754&md5=c49aa45de3817b1fea93512bb289a2ce" title="Click to view the MathML source">N≥2s. The proof is far from trivial as the two bounds are based on different combinatorial arguments.
75416301513-si7.gif">-sequences by providing a unifying framework enlightening the power and the efficiency of these methods for the study of low-discrepancy sequences. In this context, we obtain new improved explicit bounds for the star-discrepancy of these sequences, showing in most cases a better behavior than preceding ones in the non-asymptotic regime. Theoretical comparisons of discrepancy bounds in the non-asymptotic regime are much more difficult to achieve than in the asymptotic regime, where results exist to compare the leading constants 75416301513&_mathId=si15.gif&_user=111111111&_pii=S0378475416301513&_rdoc=1&_issn=03784754&md5=6fd84028d912d1bf8fac27b023107eb4" title="Click to view the MathML source">cs. Hence in this paper we mostly proceed via numerical comparisons to compare bounds. But in the case of 75416301513&_mathId=si1.gif&_user=111111111&_pii=S0378475416301513&_rdoc=1&_issn=03784754&md5=ded6e8dfc41c5a3cb2007a381c136ffb" title="Click to view the MathML source">(t,s)-sequences in base 2, we are able to compare two discrepancy bounds and prove that one is demonstrably better than the other for any 75416301513&_mathId=si17.gif&_user=111111111&_pii=S0378475416301513&_rdoc=1&_issn=03784754&md5=c49aa45de3817b1fea93512bb289a2ce" title="Click to view the MathML source">N≥2s. The proof is far from trivial as the two bounds are based on different combinatorial arguments.