Lattice methods for multiple integration: theory, error analysis and examples
SIAM Journal on Numerical Analysis
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Implementation of a lattice method for numerical multiple integration
ACM Transactions on Mathematical Software (TOMS)
Algorithm 659: Implementing Sobol's quasirandom sequence generator
ACM Transactions on Mathematical Software (TOMS)
Computational investigations of low-discrepancy sequences
ACM Transactions on Mathematical Software (TOMS)
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Monte Carlo and quasi Monte Carlo (ie using low discrepancy sequences) methods (Bratley & Fox 1988, Joe & Sloan 1993, Krommer & Ueberhuber 1994, Lyness 1989, Niederreiter 1978, Sloan & Kachoyan 1987) are used to approximate an integral by the average value of function samples:[EQUATION 1]where v is the volume of integration (taken here to be the unit multi-dimensional cube) and x is a vector with an element for each of the dimensions of the multidimensional space. In the case of Monte Carlo the points xp are chosen at random, while in quasi Monte Carlo the points are chosen to cover the integration volume as uniformly as possible. For numerical integration over a large number of dimensions these two techniques are often the only methods available.