Summary
We develop an algorithm for the construction of randomly shifted rank-1 lattice rules in d-dimensional weighted Sobolev spaces with a significantly reduced construction cost. The results shown here are an extension of earlier results by the present authors. In this new algorithm, the number of quadrature points n is a product of r distinct prime numbers p 1,…,p r. This allows us to reduce the construction cost to O(n(p 1 + … +p r)d 2), which represents a significant reduction, especially for large n. The constructed rules achieve a worst-case error bound with a rate of convergence of O(n(p 1 + δ p -1/22 ... p -1/2 r ) for any δ > 0. Numerical experiments were carried out for r = 2, 3, 4 and 5. The results demonstrate that it can be advantageous to choose n as a product of up to 5 primes.
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© 2004 Springer-Verlag Berlin Heidelberg
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Dick, J., Kuo, F.Y. (2004). Constructing Good Lattice Rules with Millions of Points. In: Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2002. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18743-8_10
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DOI: https://doi.org/10.1007/978-3-642-18743-8_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20466-4
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