1. 1 Everyday Statistics in Monte Carlo Shielding Calculations  One Key Statistics: ERROR, and why it can’t tell the whole story  Biased Sampling vs. Random Sampling

2. 2 What is a Monte Carlo Calculation?  Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results.

3. 3 Warning Monte Carlo Statistics will help us with answer how precisely we answered our question, but not how accurate our model is.

4. 4 Bottom Line in an MCNP output tally 2 tally 2 nps mean error vov slope fom nps mean error vov slope fom 512000 9.7768E-04 0.0010 0.0002 1.5 519843 512000 9.7768E-04 0.0010 0.0002 1.5 519843 nps = number of starting particles run mean = result of the tally error = standard deviation / mean vov = variance of the variance slope = slope = Pareto slope of the history score probability density function fom = figure of merit

5. 5 MCNP Details  MCNP = Monte Carlo N-Particle Code, developed at Los Alamos National Laboratory since the 1940’s  Actual MCNP outputs contain a lot of detailed data.

6. 6 Example Monte Carlo (Not MCNP) Run Sampling photons from Am-241 There are 153 photons Sample strategy – random number from 1 to 153 identifies the photon A weighting factor (a very important statistic) is used to adjust for probabilities of these photons, the lowest at 5.5E-10.

7. 7 Example Monte Carlo (Not MCNP) Run  This is biased sampling because each photon is sampled uniformly, without regard to its probability.

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9. 9 Example Pseudocode Assign a starting value for “dose.” Start a loop. Select a random integer from 1 to 153. Use the random number to select one of the photons. Multiply the photon by its weight. Add this sum to: current dose estimate * number of previous runs. Divide this by the number of current runs. End loop

10. 10 MCNP includes a lot of operations, such as:   Start a source particle (energy, direction);   Find the distance to the next boundary, cross the surface and enter the next cell;   Find the total photon cross section and process photon collisions producing electrons as appropriate;   Follow electron tracks;   Process tallies.

11. 11 In our demo, we are only going to:   Start a source particle.   Process tallies.

12. 12 Example Run -> Switch to live R presentation Switch to live R presentation <- The following slides are samples of the live presentation.

13. 13 First 200 photons

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17. 17 Up to 400

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20. 20 Up to 10,000

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22. 22 In a well-behaved Monte Carlo run, expect the error to decrease as the square of the number of samples increases. For example to divide error by 2, multiply samples by 4.

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24. 24 npsDose MC Errorvov|Error| from True 2000.02520.24050.01200.2267 4000.02440.26630.01320.2510 1,0000.02290.21440.00680.2975 10,0000.03360.13350.00130.0319 100,0000.03330.05340.00010.0218 799,4850.032579n/a 0.0004

25. 25 Time to Evaluate the Sampling Bias  Did it help or hurt our statistics to bias the sampling?  In the slides that follow, we compare unbiased sampling to biased sampling for two cases.

26. 26 Comparison to an MCNP run  Simple Model: point source in vacuum.  Tally at a sphere in vacuum.  This is very much like our R model.  Later, we add a twist: a steel shield.

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29. 29 For the Simple Case…  The random sample looks good. All statistical checks were passed.  But if you look at the output in detail…  Many of the low probability particles were not sampled at all.  We ran 10,000,000 particles, but that wasn’t enough to ensure we sampled all the particles.

30. 30 Now Add ½” Steel

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33. 33 Summary Statistics Biased Mean error vov slope fom 9.97E-06 0.0028 0.0001 10 3768 Unbiased Mean error vov slope fom 9.92E-06 0.053 0.0079 10 12

34. 34 Conclusions  How you sample makes a difference. But it depends on the problem what the preferred sampling will be.  MCNP Summary Statistics are a helpful guide, but they do not tell the whole story.