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The variance of randomly generated points within a unit square can be reduced through a stratification process. In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates obtained for a given simulation or computational effort. [1]
It allows for the efficient computation of gradients through random variables, enabling the optimization of parametric probability models using stochastic gradient descent, and the variance reduction of estimators. It was developed in the 1980s in operations research, under the name of "pathwise gradients", or "stochastic gradients".
Importance sampling is a variance reduction technique that can be used in the Monte Carlo method.The idea behind importance sampling is that certain values of the input random variables in a simulation have more impact on the parameter being estimated than others.
The control variates method is a variance reduction technique used in Monte Carlo methods. It exploits information about the errors in estimates of known quantities ...
The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path {, …,} to also take {, …,}.The advantage of this technique is twofold: it reduces the number of normal samples to be taken to generate N paths, and it reduces the variance of the sample paths, improving the precision.
Schuëller & Pradlwarter [10] reports the performance of subset simulation (and other variance-reduction techniques) in a set of stochastic mechanics benchmark problems. Chapter 4 of Phoon [11] discusses the application of subset simulation (and other Monte Carlo methods) to geotechnical engineering problems.
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In order to recover our ability to analyze and estimate the variance, we can randomize the method (see randomization for the general idea). The resulting method is called the randomized quasi-Monte Carlo method and can be also viewed as a variance reduction technique for the standard Monte Carlo method. [5]