What is quasi-Monte Carlo?
What is quasi-Monte Carlo?
Embedded Files
Monte Carlo (MC) methods are a useful numerical approximation strategy to estimate expectations, or equivalently, the integral of a multivariate function. One uses the sample mean of N randomly sampled function values as the estimate, the error of which converges at a rate of 1 over square root N.
Quasi-Monte Carlo (QMC) methods are used in place of simple MC in an attempt to improve accuracy in numerical methods by replacing the MC random sampling with low-discrepancy or space-filling sampling locations. Some examples of classical low-discrepancy sequences are digital nets and lattices. To the right is an example of rank-1 lattice.
(Two dimensional Rank-1 Lattice Construction)
(Two dimensional projections of a six dimensional Sobol' sequence construction)
Most QMC constructions are studied and evaluated based on their asymptotic behaviour as N approaches infinity. However, few methods address finding optimal node configurations for specific N and d which can be critical in practical applications where each function evaluation is quite costly. Of course, one possible strategy is to truncate asymptotically optimal sequences to the desired n, however, as discussed above many of the standard low-discrepancy constructions have a preferred number of nodes with truncations leading to imbalances in equidistribution. We can often do better optimizing directly for the target N and d.
The optimization of low-discrepancy nodes is precisely the focus of my recent research.
More information can be found on quasi-Monte Carlo methods in our recent tutorial paper, aimed at students and researchers that are interested in learning more about QMC and slides from a talk in October 2025.
Page updated
Google Sites
Report abuse