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Constructing Good Lattice Rules with Millions of Points

In: Monte Carlo and Quasi-Monte Carlo Methods 2002

Author

Listed:
  • Josef Dick

    (University of New South Wales, School of Mathematics)

  • Frances Y. Kuo

    (University of New South Wales, School of Mathematics)

Abstract

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 2 -1/2 ... p r -1/2 ) 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.

Suggested Citation

  • Josef Dick & Frances Y. Kuo, 2004. "Constructing Good Lattice Rules with Millions of Points," Springer Books, in: Harald Niederreiter (ed.), Monte Carlo and Quasi-Monte Carlo Methods 2002, pages 181-197, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-18743-8_10
    DOI: 10.1007/978-3-642-18743-8_10
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