Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

458 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsnumbers in Table 7.2 yield the parameter values 0.09, 0.125, and 0.168 for survival probabilities0.3, 0.5, and 0.7, respectively. For c = 0.9, the value of w thmay vary between0.12 and 0.31 (corresponding to n - 19 and n = 11, respectively) without considerablechange of the efficiency.Nevertheless, it should be emphasized here, too, that the above results should be consideredmore as illustrations of an optimization method than as a definite recipe for determiningthe Russian roulette parameter. It has also been seen in the simplified example abovethat, in certain cases, the efficiency of a game with Russian roulette is very sensitive to thechoice of the parameter. This may be even more the case in more complicated problems.Competent practitioners say 26about the Russian roulette parameter that "It is very problemdependent and its setting is an art." A systematic study of this common Monte Carlo toolis very much needed, and such a study may lead to surprising results, astonishing even forold practitioners.I). OPTIMIZATION BY DIRECT STATISTICAL APPROACHThe approximate analytical optimization models so far investigated in this Chapter wereseen to reflect certain characteristic properties of a real simulation and provided us withuseful qualitative information on the efficiency-increasing schemes considered. Nevertheless,these models are so oversimplified that the quantitative results they produce may not beconsidered realistic.The loss of information caused by the application of simplified models can be compensatedfor to some extent by the application of parameters determined from numerical experiments.In the introduction, this procedure was called the direct statistical approach. Inthis section, we illustrate the method in two simple cases; more elaborate models will begiven in subsequent Chapters.Let us first consider the optimization of a single-surface geometrical splitting procedureby the direct statistical approach. Assume that particles are started from a "source region"and we wish to estimate the number of particles that reach a "detector region". For thesake of simplicity, we suppose that the two regions have no common part and there existsa surface that completely separates the two regions in such a way that any particle that startsfrom the source region and reaches the detector region must cross the surface. We define asplitting procedure in which a particle is split into n fragments when it crosses the surfacefor the first time, but no repeated splitting is played in case of a second or further crossing.Optimization of the procedure consists of selecting an n value that maximizes the efficiencyof the game. The efficiency will be formulated in terms of "average probabilities" whichdescribe a "typical particle" and which will then be defined through experimental values.Let p, be the probability that a typical source particle reaches the splitting surface andlet p 2be the probability that a typical fragment starting from the splitting surface reachesthe detector region. Let W 1denote the average weight of a unit-weight starter when it reachesthe splitting surface and W 2denote the average weight of a fragment in the detector regionif its weight was unity at the splitting surface. Then the quantity to be estimated can beexpressed asR -- P 1P 2W 1W 2(7.39)Now, in an n-for-one splitting, the probability that k out of the n fragments of a starterreaches the detector region after a splitting isQ K= P,©PL(I ~ P 2R Kand since the average weight of a fragment is w,/n, its average weight in the detector region