Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

90 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsThenN, =NJand the estimate (4.19) becomes:E h(x„)(4.20)In the actual realization of (4.20) there is a possibility to use the same random numbersequence (i = 1,2, . . . ,N/J) for the selection of the x Lj-s for j = 1,2, . . . ,J, howeverthis method — in spite of the fact that it needs fewer random numbers — results in generallyworse efficiency than the use of N independent samples.Systematic sampling always reduces the variance from that obtained by the straightforwardmethod. The variance is decreased by J 2in the equal probability domain division, inthe general case the estimation of the gain is more complicated. An exhaustive analysis ofthe method was given by Kahn. 14Just for comparison, the domain can be imagined to be sliced into sub-domains also inthe straightforward sampling, even if this slicing is actually not used. From this point ofview, the difference between the estimates in Equations (4.8) and (4.19) is that the precomputedNj values in case of systematic sampling are replaced by randomly selected values ifthe straightforward sampling is used.The use of systematic sampling is always recommended if it does not need significantextra work.I). QUOTA SAMPLINGThe basic concept of quota sampling — or stratified sampling as it is alternatively called— is to mix the ideas of the importance and systematic samplings. The Y domain is againcut into subdomains, however the number of samples to be taken from each subdomain isselected such as to obtain a minimum — or, in practice, near-minimum — variance. Regionsof large variances should be sampled more frequently.The rules of slicing the F domain are the same as given by Equations (4.17) and (4.18)however now the N-s are to be selected as to obtain a minimum variance.In quota sampling the first equality of Equation (4.19) is still applicable and the estimateis now:(4.21)ifthe contribution from the j-th subdomain.The unbiasedness of the estimator in Equation (4.21) can be proved exactly in the samemanner as applied in the proof of Theorem 4.5.