Importance sampling changes the second part of Equation (17.42) into pf =

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f (c, ) h(c, )dcd h(c, )

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(17.44)

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which leads to the Monte Carlo approximation pf 1 n

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f (ci , i ) h(ci , i )

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(17.45)

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where Ji is as de ned for the basic Monte Carlo method. This leads to the following computational procedure: 1. Establish n = the number of simulation points to be used. 2. For each point i generate a pair of values ci and i from the distribution h(ci , i ). Note that this is not f (ci , i ) and does not include the correlation. 3. If M(ci , i ) > 0, skip to the next i. If M(ci , i ) 0, calculate f (ci , i )/ h(ci , i ), add the result to a running sum, and go to the next i. 4. After all n points have been evaluated, the estimate of pf is the running sum divided by n. The correct probability of failure for the parameters in Example 17.1 is 0.0362. Both the basic Monte Carlo simulation and the importance-sampled simulation were run for n = 10, 100, and 1000, and each simulation was repeated ten times. Figure 17.9 shows the results. There are two vertical lines above each value of n. The right one represents the results for the basic simulation, and the left one corresponds to the importance sampled simulation. The top of the line is the largest estimate of pf , the bottom is the smallest value, and the horizontal line near the middle of the vertical line represents the mean result. It is not surprising that results for basic simulation with n = 10 are poor, but the mean result for the importance sampled simulation with n = 10 is remarkably good. The results

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MONTE CARLO SIMULATION METHODS

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Convergence of Monte Carlo Simulation

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0.1 0.08 0.06 0.04 0.02 0

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Probability of Failure

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n = 10

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n = 100

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n = 1000

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exact

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Simulations (left = importance sampled)

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Figure 17.9 Probability of failure of vertical cut 10 m high evaluated by Monte Carlo simulation with and without importance sampling with different numbers of simulations. Results on the left are importance sampled. Lines span between the smallest and largest values from 10 sets of simulations; line near middle is average of 10 results.

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get better as n increases, and in each case the importance-sampled results are signi cantly superior. When n = 1000, the results with importance sampling are nearly exact. These results demonstrate the particular usefulness of importance sampling when the probability to be calculated is small. The convergence could be improved further by choosing a better importance function. Hasofer and Wang (1993) propose a more complicated function that is exponentially distributed in a direction normal to the failure criterion and normally distributed in the remaining directions. While this would probably give better results, it is more complicated, and the present results demonstrate the effectiveness of importance sampling even with a relatively crude importance function. Another point to be emphasized is that the same calculation could be done using the factor of safety instead of the margin of safety. The criterion that F S 1 is identical to the requirement that M 0. Thus, the same center and form for the importance function are de ned, and the computational results are identical.

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Antithetic Sampling

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Antithetic sampling can be useful when there are reasons to believe that different senses of deviation from the mean values of the variates will have different effects on the value of variable to be integrated. Suppose ZA and ZB are two unbiased estimators of I . Then Z = 1 (ZA + zB ) 2 is an unbiased estimator of I . The variance of Z is Var(Z) = 1 [Var(ZA ) + Var(ZB ) + 2 Cov (ZA , ZB )] 4 (17.47) (17.46)

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