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Lesson 11: Go over take-home + miscellaneous topics. This class is scheduled to be a catch-up, so we will go after the following: Discussion of the final exam ( takehome ) Point flux estimator FOM as measure of convergence of deep penetration shielding problem MCNP 10 statistical checks.
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Lesson 11: Go over take-home + miscellaneous topics • This class is scheduled to be a catch-up, so we will go after the following: • Discussion of the final exam (takehome) • Point flux estimator • FOM as measure of convergence of deep penetration shielding problem • MCNP 10 statistical checks
Point Flux Estimators • It is impossible to determine the flux at a point using traditional Monte Carlo: • For collision estimator, a collision would have to occur AT the point desired • For path length estimator, the particle would have to pass THROUGH the point • Both are impossible • Furthermore, tallies have to be integrals • Therefore: We must reduce the point flux to an integral based on the collision rate variable • We treat an emerging particle like an point, monoenergetic, anisotropic beam source
Point Flux Estimators (2) • The c(...) represents “emerging particles”, which includes both original source particles and particles coming out of a scattering collision • Basically this means that we estimate the
Point Flux Estimators (2) = • The c(...) represents “emerging particles”, which includes both original source particles and particles coming out of a scattering collision • Basically this means that we estimate the flux at a new point from particles emerging ANYWHERE else in the problem • In practice, we do this AFTER we have chosen the source point and new energy, but BEFORE we have chosen the new direction
Figure of Merit (FOM) • Of practical significance beyond considerations of variance is the idea of Figure of Merit (FOM) which gives us a relative measure of Monte Carlo efficiency • Theoretical basis: where T=computer time • Of practical concern: • We invert this (so that bigger is better):
Use of FOM In our previous (brief) discussion of FOM, I mentioned two uses: • Relative metric to compare two computers or new version of code vs. old version, etc. • Relative metric of gain from a “variance reduction” process There is actually a third: • Good measure of whether a problem has reached statistical stability (i.e., is delivering the right answer)
MCNP 10 statistical check #1 • Coding of an “eyeball” trick that confirms that the answers—as time goes by—are really bouncing around a central value and not trending up or down (or both)
MCNP 10 statistical check #2 • Biggest problem of interpreting the answer for a deep penetration—long running—MCNP calculation: • You are so hungry for an answer after hours (or days) of waiting for a solution that you believe MCNP when it tells you (for example) that it knows the answer within 20% or 40% • This is a complete fiction: An error of 100% (1.00) means that ONE particle has scored. • So this translates into needing at least 100 contributing particles before we “believe”
MCNP 10 statistical check #3 • Translates into the requirement that no high-scoring “rogue” particle has disrupted the answer during the last half of the problem • Too much “settling down” is going on in the first half to apply this rule of thumbding at least 100 contributing particles before we “believe”
MCNP 10 statistical check #4 • We know theoretically that the “inverse square root of N” rule applies to a stable statistical process • Meaning that we are sampling meaningful particles often enough that we should not expect future surprises
MCNP 10 statistical check #5 • This is the equivalent of the 0.1 for the answer itself • “Variance of the variance” is just a mind-bending expression that says our ESTIMATE of the variance (and then the standard deviation) itself settles into an answer in time.
MCNP 10 statistical check #6 • Equivalent to #3 for the variance (standard deviation)
MCNP 10 statistical check #7 • Equivalent to #4 for the variance. Notice that it converges as 1/N instead of 1/sqrt(N) • This means, of course, that we know it better sooner
MCNP 10 statistical check #8 • FOM (as discussed earlier) is supposed to be a constant for a statistically stable process being sampled. • This is just a measure that the problem is not “drifting”
MCNP 10 statistical check #9 • This goes a little beyond the previous measure, as another coding of the “eyeballing” of drift in your solution
MCNP 10 statistical check #10 • This is the hardest to grasp of the 10 • It has to do with the “upper end” of the distribution of your scores • The idea is that your MC process viewed as a whole is sampling of a distribution. • This is a guess of whether the distribution we are sampling actually HAS a mean or not • What is magic about the slope of 3 (actually -3)?
