Smoothness and dimension reduction in quasi-Monte Carlo methods

B. Moskowitz, Russel Caflisch

Research output: Contribution to journalArticle

Abstract

Monte Carlo integration using quasirandom sequences has theoretical error bounds of size O(N-1 logd N) in dimension d, as opposed to the error of size O(N-1/2) for random or pseudorandom sequences. In practice, however, this improved performance for quasirandom sequences is often not observed. The degradation of performance is due to discontinuity or lack of smoothness in the integrand and to large dimension of the domain of integration, both of which often occur in Monte Carlo methods. In this paper, modified Monte Carlo methods are developed, using smoothing and dimension reduction, so that the convergence rate of nearly O(N-1) is regained. The standard rejection method, as used in importance sampling, involves discontinuities, corresponding to the decision to accept or reject. A smoothed rejection method, as well as a method of weighted uniform sampling, is formulated below and found to have error size of almost O(N-1) in quasi-Monte Carlo. Quasi-Monte Carlo evaluation of Feynman-Kac path integrals involves high dimension, one dimension for each discrete time interval. Through an alternative discretization, the effective dimension of the integration domain is drastically reduced, so that the error size close to O(N-1) is again regained.

Original languageEnglish (US)
Pages (from-to)37-54
Number of pages18
JournalMathematical and Computer Modelling
Volume23
Issue number8-9
StatePublished - 1996

Fingerprint

Quasi-Monte Carlo Methods
Dimension Reduction
Smoothness
Monte Carlo methods
Rejection Method
Quasi-Monte Carlo
Monte Carlo method
Discontinuity
Effective Dimension
Monte Carlo Integration
Pseudorandom Sequence
Importance sampling
Random Sequence
Importance Sampling
Integrand
Curvilinear integral
One Dimension
Error Bounds
Higher Dimensions
Smoothing

Keywords

  • Acceptance-rejection
  • Feynman-Kac
  • Monte Carlo
  • Quasirandom
  • Weighted uniform sampling

ASJC Scopus subject areas

  • Modeling and Simulation
  • Computer Science Applications

Cite this

Smoothness and dimension reduction in quasi-Monte Carlo methods. / Moskowitz, B.; Caflisch, Russel.

In: Mathematical and Computer Modelling, Vol. 23, No. 8-9, 1996, p. 37-54.

Research output: Contribution to journalArticle

Moskowitz, B & Caflisch, R 1996, 'Smoothness and dimension reduction in quasi-Monte Carlo methods', Mathematical and Computer Modelling, vol. 23, no. 8-9, pp. 37-54.
Moskowitz, B. ; Caflisch, Russel. / Smoothness and dimension reduction in quasi-Monte Carlo methods. In: Mathematical and Computer Modelling. 1996 ; Vol. 23, No. 8-9. pp. 37-54.
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