Editing Quasi-Monte Carlo method (section)

{{Use American English|date=January 2019}}{{Short description|Numerical integration process}}{{multiple image
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   | footer    = 256 points from a pseudorandom number source, and Sobol sequence (red=1,..,10, blue=11,..,100, green=101,..,256). Points from Sobol sequence are more evenly distributed.
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   | caption1  = [[Pseudorandom]] sequence
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   | caption2  = A [[Sobol sequence]] of [[Low-discrepancy sequence|low-discrepancy]] quasi-random numbers, showing the first 10 (red), 100 (red+blue) and 256 (red+blue+green) points from the sequence.
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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 sequence]]s (also called quasi-random sequences or sub-random sequences) to achieve [[variance reduction]]. 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 ''x''<sub>1</sub>, ..., ''x''<sub>''N''</sub>:

:<math> \int_{[0,1]^s} f(u)\,{\rm d}u \approx \frac{1}{N}\,\sum_{i=1}^N f(x_i). </math>

Since we are integrating over the ''s''-dimensional [[unit cube]], each ''x''<sub>''i''</sub> is a vector of ''s'' elements. The difference between quasi-Monte Carlo and Monte Carlo is the way the ''x''<sub>''i''</sub> 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(1/''N''), whereas the rate for the Monte Carlo method is O(''N''<sup>−0.5</sup>).<ref name="asmunssen_glynn_book">Søren Asmussen and Peter W. Glynn, ''Stochastic Simulation: Algorithms and Analysis'', Springer, 2007, 476 pages</ref>

The Quasi-Monte Carlo method recently became popular in the area of [[mathematical finance]] or [[computational finance]].<ref name="asmunssen_glynn_book" /> 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.