Stochastic counterpart approximation

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Due to the complexity of numerical integration and the exponential increase in sample size with the increase of the random variables, we employ an approximation scheme know as the sample average approximation (SAA) method, also known as stochastic counterpart. The SAA problem can be written as ... 

This approach to the numerical solution of stochastic optimization problems is a natural outgrowth of the Monte Carlo method of estimation of the expected value of a random function. The method is known by various names, and it is difircirlt to point out who was the first to suggest this approach. In the recent literature a variant of this method, based on the likelihood ratio estimator g (x), was suggested in Rubinstein and Shapiro (1990) imder the name stochastic counterpart method (tilso see Rubinstein and Shapiro 1993 for a thorough discussion of such a Ukelihood ratio-stunple approximation approach). In Robinson (1996) such an approach is called the sample path method. This idea can also be applied to cases in which the set x is finite, that is, to stochastic discrete optimization problems (Kleywegt and Shapiro 1999). 

For the SFE solution of stochastic boundary value problems, a mathematical theory is available that is in many aspects comparable to its deterministic counterpart, the finite element method. However, for random fields with short correlation lengths (requiring a high number M of random variables), the solution methods become inefficient, due to the series expansion of the solution. This is also tme for most other SFE approximations, be they global or local, and recourse to efficient sampling... 

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