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In numerical analysis, the quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences. This is in contrast to the regular Monte Carlo method or Monte Carlo integration, which are based on sequences of pseudorandom numbers.
Monte Carlo and quasi-Monte Carlo methods are stated in a similar way.The problem is to approximate the integral of a function f as the average of the function evaluated at a set of points x1,..., xN:
Since we are integrating over the s-dimensional unit cube, each xi is a vector of s elements. The difference between quasi-Monte Carlo and Monte Carlo is the way the xi are chosen. Quasi-Monte Carlo uses a low-discrepancy sequence such as the Halton sequence, the Sobol sequence, or the Faure sequence, whereas Monte Carlo uses a pseudorandom sequence. The advantage of using low-discrepancy sequences is a faster rate of convergence. Quasi-Monte Carlo has a rate of convergence close to O, whereas the rate for the Monte Carlo method is O.
The Quasi-Monte Carlo method recently became popular in the area of mathematical finance or computational finance. In these areas, high-dimensional numerical integrals, where the integral should be evaluated within a threshold ε, occur frequently. Hence, the Monte Carlo method and the quasi-Monte Carlo method are beneficial in these situations.