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How Many Random Bits Do We Need for Monte Carlo Integration?

In: Monte Carlo and Quasi-Monte Carlo Methods 2002

Author

Listed:
  • Stefan Heinrich

    (FB Informatik, Universität Kaiserslautern)

  • Erich Novak

    (Mathematisches Institut, Universität Jena)

  • Harald Pfeiffer

    (Mathematisches Institut, Universität Jena)

Abstract

Summary We study Monte Carlo methods (randomized algorithms) that use only a small number of random bits instead of more general random numbers for the computation of sums and integrals. To approximate N -1 ∑ i=0 N-1 f i for f ∈ R N ,the classical Monte Carlo method uses n function values, that is, coordinates of f, and n random numbers. Our method gives the same error with only 2[log2 N] random bits, independently of n. To approximate ∫[0, 1] d f (x) dx for f from a Sobolev space, the classical Monte Carlo method uses n function values and d.n random numbers. We present a method with the optimal order of convergence that uses only at most (2 + d) log2 n random bits.

Suggested Citation

  • Stefan Heinrich & Erich Novak & Harald Pfeiffer, 2004. "How Many Random Bits Do We Need for Monte Carlo Integration?," Springer Books, in: Harald Niederreiter (ed.), Monte Carlo and Quasi-Monte Carlo Methods 2002, pages 27-49, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-18743-8_2
    DOI: 10.1007/978-3-642-18743-8_2
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