Variance reduction

If the function being integrated does not fluctuate too much in the interval of integration, and does not differ much from the average value, then the standard Monte Carlo mean-value method should work well with a reasonable number of points. Otherwise, we will find that the variance is very large, meaning that some points will make small contributions, while others will make large contributions to the integral. If this is the case, the algorithm will be very inefficient. The method can be improves by splitting the function $f(x)$ in two $f(x)=f_1(x)+f_2(x)$, such that the integral of $f_1(x)$ is known, and $f_2(x)$ as a small variance. The ``variance reduction'' technique, consists then in evaluating the integral of $f_2(x)$ to obtain:

$$\int _a^b{f(x)dx}=\int _a^b {f_1(x)dx} + \int _a^b{f_2(x)dx} = \int _a^b{f_1(x)dx}+J.$$