eldo_user

Monte Carlo Analysis
Importance Sampling Monte Carlo Examples
RTOL
Table 11-1. Speedup of ISMC Over Standard Monte Carlo
Effective Relative
Precision
Probability
Estimator
Number of
Runs
Speedup
20% 10.80% 9.6×10 −6 7040 4.9×10 3
10% 7.87% 9.65×10 −6 8060 8.0×10 3
5% 4.88% 9.11×10 −6 11120 1.6×10 4
1% 1.83% 9.051×10 −6 53960 2.3×10 4
In general, the number of runs does not significantly decrease as the precision increases
(compare 10% and 20% RTOL in Table 11-1). This is because the ISMC method uses batches
of runs, and a whole batch must be completed before the accuracy can be determined. However,
decreasing the batch size would have a negative impact on the rate of convergence and the
robustness of the results.
Now compare 10% and 1% RTOL in Table 11-1. It is known that to achieve a single extra digit
of precision, the number of standard Monte Carlo runs must increase by a factor of 100. For
effective precision, the results in Table 11-1 (7.9% precision with 8060 runs and 1.8% precision
with 53960 runs) give the following reduction in variance:
This speedup remains moderate with respect to one obtained using the standard Monte Carlo
method. It also indicates that sampling for a high precision is (still) difficult. The default value
of 10% was chosen from this practical point of view.
The following .EXTRACT MC … MCROB() statement specifies a rare events simulation:
.EXTRACT MC LABEL=prob_rare MCPROB(write_delay_qx, GE, 63.40)
The following results are obtained using the ISMC method:
Lower Pr. Value Upper CV(%) #Runs Speedup
Tail / Complementary
--------------------------------------------------------------------------
3.30e-09 3.65e-09/1.00e+00 4.00e-09 9.66 24340 4.6e+06
The failure probability is obtained with a precision of 9.66% using 24340 Monte Carlo runs,
yielding a variance reduction factor of over 4 million. The standard Monte Carlo method would
require over 12 billion runs to obtain comparable results.
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Eldo® User's Manual, 15.3