Variance reduction techniques

Wilson JR. 1984. Variance reduction techniques for digital simulation. Am J Math Manage Sci 4 277-312. 

The goal of variance reduction is to decrease the error of a point estimate, which should lead to a smaller estimated standard error or a narrower confidence interval (see Section 8 for a discussion of measures of error). This section describes the variance reduction technique known as common random numbers (CRN), which is useful for reducing the error in comparing the expected performance of two or more systems. The presentation is based on Nelson (1987). 

Variance Reduction Techniques Importance Sampling (IS), Line Sampling (LS), etc. 

In spite of the arguments discussed above, the concept of design point in reUabil-ity analysis should not be discarded. Recent advances in the development of variance reduction techniques (Au 2008) have shown that the design point (associated with uncertainty in excitation) can be rather useful when interpreted as the excitation with minimum energy capable of driving the structural reponse towards a prescribed threshold level. But how to estimate the design point efficiently still remains an open issue, except for a small class of reliability problems, where exact or approximate solutions have been proposed. 

Among variance reduction techniques the importance sampling technique is most frequently applied in structural reliability analysis e.g. in [1,3,10-12] whereas other techniques e.g. stratified sampling [13], Latin hypercube sampling [14,15] and antithetic variates [11], also proved to be very powerful tools for structural reliability analyses. The concepts of these techniques will be discussed briefly below. 

Among the variance reduction techniques for simulation the Importance sampling procedure proved to be most advantageous. The importance Sampling Rrocedure Using Resign Points (ISPUD) - for which an elaborate software environment is already available - Is shown to be very flexible in practical application. Its accuracy and efficiency is quite satisfactory. 

There are many successful ways to increase the efficiency of the Monte Carlo method, in particular by decreasing the dispersion, and these are described in several textbooks. Numerous variance reduction techniques, such as splitting/Russian roulette, weight windows, and the exponential transformation method have been proposed to improve the efficiency of Monte Carlo transport calculations. 

In this paper, in order to decrease the uncertainty of sampling, the variance reduction techniques have been used. Variance reduction techniques are methods that attempt to reduce the variance, i.e., the dispersion associated with the variations, of the parameter being evaluated. 

Smith, D.T., Johnson, B.W., Andrianos, N., Profeta HI., J.A. A variance-reduction technique via fault-expansion for fault-coverage estimation. IEEE TR 46(3), 366-374 (1997)... 

The Girsanov Transformation-Based Variance Reduction Technique... 

For linear systems and linear performance functions, the suboptimal control force can be obtained as a function of impulse response functions reversed in time. Also from Eq. 15, it can be noted that the evaluation of the correction process, also known as the Radon-Nikodym derivative, is independent of the mathematical model for the structure under study. Thus, an acceptable choice for the control force can be made solely based on experimental techniques, and the estimator for the reliability can be deduced without taking further recourse to mathematical model for the structure tmder study. This permits the application of the Girsanov transformation-based variance reduction technique in the experimental study of time variant rehability of complex structural systems which are difficult to model mathematically (Sundar and Manohar 2014a). 

Different variance reduction strategies can be combined into a single framework so as to leverage advantages associated with each one of them. Thus, for example, the subset simulation method and the Girsanov transformation-based method can be combined in such a way that subset simulations handle the uncertainties associated with parameters, and the Girsanov transformation takes care of the random excitations (Sundar and Manohar 2014b). Performance of few simulation-based variance reduction techniques with respect to few benchmark problems has been documented by Schueller and Pradlwarter (2007). 

The direct MCS examined in section Response Variability of Stochastic Systems becomes inefficient for the solutimi of reliability problems where a large number of low-probability realizations in the failure domain must be produced. In order to alleviate this problem without deteriorating the accuracy of the solutimi, numerous variants of this approach have been developed. An important class of improved MCS is variance reduction techniques where the generation of samples of the basic random variables is controlled in an efficient way. 

Recall that Eq. 14 provided an estimate for the variance Pf. Using the so-called variance reduction techniques, it is possible to reduce this variance and thereby obtain an improved estimate of Pf. Such techniques are called variance reduction techniques (Kalos and Whitlock 2008) and are commonly used while estimating Pf. One such technique is popularly called the conditional expectation method in this method, a control variable is selected and the variance of Pf is reduced by removing the random fluctuations of this control variable which was not conditioned. In another technique, popularly known as the technique of antithetic variates, negative correlation is purposefully induced between successive samples to decrease the variance of the estimated mean value. It is also common to use the technique of antithetic variates in combination with the conditional expectation method (Haidar and Mahadevan 2000). 

---