Good lattice rules based on the general weighted star discrepancy

Authors:
Vasile Sinescu and Stephen Joe

Journal:
Math. Comp. **76** (2007), 989-1004

MSC (2000):
Primary 65D30, 65D32; Secondary 11K38

DOI:
https://doi.org/10.1090/S0025-5718-06-01943-0

Published electronically:
December 12, 2006

MathSciNet review:
2291846

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Abstract: We study the problem of constructing rank- lattice rules which have good bounds on the ``weighted star discrepancy''. Here the non-negative weights are general weights rather than the product weights considered in most earlier works. In order to show the existence of such good lattice rules, we use an averaging argument, and a similar argument is used later to prove that these lattice rules may be obtained using a component-by-component (CBC) construction of the generating vector. Under appropriate conditions on the weights, these lattice rules satisfy strong tractability bounds on the weighted star discrepancy. Particular classes of weights known as ``order-dependent'' and ``finite-order'' weights are then considered and we show that the cost of the construction can be very much reduced for these two classes of weights.

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Additional Information

**Vasile Sinescu**

Affiliation:
Department of Mathematics, University of Waikato, Hamilton, New Zealand

Email:
vs27@waikato.ac.nz

**Stephen Joe**

Affiliation:
Department of Mathematics, University of Waikato, Hamilton, New Zealand

Email:
stephenj@math.waikato.ac.nz

DOI:
https://doi.org/10.1090/S0025-5718-06-01943-0

Keywords:
Rank-$1$ lattice rules,
weighted star discrepancy,
component-by-component construction

Received by editor(s):
August 23, 2005

Received by editor(s) in revised form:
April 20, 2006

Published electronically:
December 12, 2006

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.