A publication of the Advanced Simulation & Computing Division, NA-121.2, NNSA Defense Programs
December 2008
NA-ASC-500-08—Issue 9
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Protection Against Erroneous Monte Carlo Calculations
Note that Monte Carlo calculations are statistical by nature. Every time a Monte Carlo calculation of a problem is done, a different answer (estimated mean) for the problem results. Without knowledge of a confidence interval, “the” answer from a Monte Carlo calculation is almost meaningless. It is like a political poll, the answer depends on the random choice of the particular people polled. The average result of the poll is meaningful only if there is an estimate of the error in the poll numbers. Just as no sophisticated political campaign would base actions on a poll with a huge margin of error, no scientist or engineer wants to base the analysis of nuclear systems on calculations with unknown margins of error.
Monte Carlo calculations rely on the central limit theorem of statistics to produce valid confidence intervals for the estimates. The central limit theorem, in turn, requires the estimates to have finite variances. Current Monte Carlo codes (e.g., MCNP) are not protected against attempting calculations having infinite variance, so current practice is simply to assume that the variance is finite, unless there is some obvious empirical evidence to the contrary. The necessity of this assumption means that the stated confidence intervals may not be valid in some cases.
In a paper available online in Mathematics and Computers in Simulation, Los Alamos staff R. R. Picard and T. E. Booth report mathematical proofs that a small, but extremely general, sampling modification, converts many infinite variance samplings seen in practice to finite variance samplings. Furthermore, empirical evidence suggests that this small modification, in fact, will convert all infinite variance samplings to finite variance samplings. To see the paper, go to http://dx.doi.org/10.1016/j.matcom.2008.11.014.
The figure shows how the error behaves in a slab penetration problem both with and without the sampling modification. Note the erratic error behavior without the modification and the very smooth error behavior with the modification.
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