As noted previously, the accuracy of the model output statistics and probability distribution (e. g., probability that a specified safety level will be exceeded) obtained from Monte Carlo simulation is a function of the number of simulations performed. For models or problems with a large number of uncertain basic variables and for which low probabilities (<0.1) are of interest, tens of thousands of simulations may be required. Rules for determining the number of simulations required for convergence are not available, and thus replication of the Monte Carlo simulation runs for a given number of simulations is the only way to check convergence (Melching, 1995). Cheng et al. (1982) considered the convergence characteristics of Monte Carlo simulation for a simple case of Z = X3X4 — (X1 + X2), where the distributions and statistics of the variables are listed in Table 6.1. They found that failure probabilities (i. e., probability of Z < 0) down to 0.0025 could be estimated reliably with 32,000 simulations, failure probabilities down to 0.015 could be estimated reliably with 8000 simulations, and failure probabilities down to 0.2 could be estimated reliably with 1000 simulations.
Problems involving more complex system functions Z and more basic variables may require more simulations to obtain similar accuracy. For example, Melching (1992) found that 1000 simulations were adequate to estimate the mean, standard deviation, and quantiles above 0.2 for an application of the HEC-1 (U. S. Army Corps of Engineers, 1990) and RORB (Laurenson and Mein, 1985) rainfall-runoff models and that 10,000 simulations were needed to accurately estimate quantiles between 0.01 and 0.2. Brown and Barnwell (1987) reported that for the QUAL2E multiple-constituent (dissolved oxygen, nitrogen cycle, algae, etc.) steady-state surface water-quality model, 2000 simulations were required to obtain accurate estimates of the output standard deviation. With the computational speed of today’s computers, making even 10,000 runs is not prohibitive for simpler models. However, increased computational speed has made possible the use of computational fluid dynamics codes in three dimensions for hydrosystems design work. When such codes are applied, the variance-reduction techniques described in Sec. 6.7 may be preferred to Monte Carlo simulation.
This chapter focuses on the basic principles and applications of Monte Carlo simulations in the reliability analysis of hydrosystems engineering problems. Section 6.2 describes some basic concepts of generating random numbers, followed by discussions on the classifications of algorithms for a generation of random variates in Sec. 6.3. Algorithms for generating univariate random numbers
|
TABLE 6.1 Basic Variable Statistics and Distributions for Evaluation of Monte Carlo Simulation of Convergence
SOURCE: After Cheng et al. (1982). |
are described in Sec. 6.4 for several commonly used distribution functions. In Sec. 6.5, attention is given to algorithms that generate multivariate random numbers. As reliability assessment involves mathematical integration, Sec. 6.6 describes several Monte Carlo simulation techniques for reliability evaluation. Given that Monte Carlo simulations, in essence, are sampling techniques, they provide only estimations, which inevitably are subject to certain degrees of errors. To improve the accuracy of the Monte Carlo estimation while reducing excessive computational time, several variance-reduction techniques are discussed in Sec. 6.7. Finally, resampling techniques are described in Sec. 6.8, which allow for assessment of the uncertainty of the quantity of interest based on the available random data without having to make assumptions about the underlying probabilistic structures.
