VARIANCE REDUCTION METHODS

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If ZA and ZB are uncorrelated, then Var(Z) = 1 [Var(ZA ) + Var(ZB )] 4 (17.48)

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Therefore, if Cov (ZA , ZB ) < 0, the variance in Equation (17.47) is less than the variance when the two samples are uncorrelated, and the simulation will be more ef cient. One widely used technique for making ZA and ZB negatively correlated is to create ZA from n simulations and ZB from n complementary simulations, making a total of 2n simulations. By the complement we mean that, if X is uniformly distributed between 0 and 1, each xB is equal to the corresponding xA subtracted from 1, and, if X has a standard Normal distribution, each xB is equal to the negative of the corresponding xA . It is easy to derive the corresponding relations for other distributions. The algorithm works as follows: 1. 2. 3. 4. Perform one simulation. Compute the complementary random numbers and perform another simulation. Repeat steps 1 and 2 a total of n times. Combine the results as though this had been one simulation with 2n samples.

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Obviously, this will work only if the complementary random numbers do actually generate negatively correlated results. The following example shows what happens when this condition is satis ed and when it is not. Example 17.4 Antithetic Sampling for Culmann Failure along Single Plane The Culmann analysis applies to the failure of a slope along a single plane. The following parameters appear in the solution: H = height of slope = 10 m in this case = inclination of failure plane = 20 in this case = inclination of face of slope = 26 in this case c = cohesion, c = 5 kPa, c = 2 kPa in this case tan = tangent of friction angle, tan = 0.5774, tan = 0.1386 in this case = unit weight, = 5 kN/m3 , = 2 kN/m3 in this case All variables are uncorrelated. The mean value of tan corresponds to = 30 . The factor of safety, FS, is c+ FS = 1 H sin( ) cos tan 2 sin 1 H sin( ) sin 2 sin

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(17.49)

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and the margin of safety is M =c+ 1 H sin( ) cos (tan tan ) 2 sin (17.50)

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Since only c, , and tan are uncertain, exact values of M and M can be calculated. The results are M = 10.26 kPa and M = 4.15 kPa.

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MONTE CARLO SIMULATION METHODS

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The analysis consisted of Monte Carlo simulation of both M and FS. The basic simulations used 10, 100 and 1000 points. For each value of n (number of points), ten independent simulations were run. All the uncertain variables were assumed to be Normally distributed. Then, the simulations were repeated with the same rst n/2 points and a second group of n/2 points using the complements of these values. Mean values of M and FS were calculated for each sampling. The results are shown in Figure 17.10 for the factor of safety. The plot shows the maximum and minimum and the mean of the results for each group of simulations. Antithetic sampling clearly improves the convergence. Indeed, the results using ten points with antithetic simulation are better than those from the simulations using 1000 points. Similar results occur for the margin of safety. When the results of these simulations are used to compute the standard deviations of M or FS, the results are completely different, as Figures 17.11 shows. The results from antithetic simulation are worse than those from direct simulation. Study of Equations (17.49)

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Mean Factor of Safety for Culmann Model 2.3 2.2 2.1 2 1.9 1.8 n = 10 n = 100 Number of Simulations Basic MC Antithetic n = 1000

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Figure 17.10 Results for basic Monte Carlo simulation and simulation with antithetic sampling of mean factor of safety for Culmann single plane failure model.

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St. Dev. Factor of Safety for Culmann Model 0.6 0.55 Est. St. Dev. 0.5 0.45 0.4 0.35 0.3 n = 10 n = 100 Number of Simulations Basic MC Antithetic n = 1000

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Figure 17.11 Results for basic Monte Carlo simulation and simulation with antithetic sampling of standard deviation of factor of safety for Culmann single plane failure model.

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