Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

284 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswhere, again, M 1represents the i-th moment of the score in such histories. Assume that wewish to compare the efficiencies of a given nonanalog game and the corresponding analoggame, the latter beingE = {[M 2(P) - MT(P)]N(P)}-'Let us denotePo(P) = [M 2(P) - M 2(P)|/[M 2(P) - Mt(P)I (5.361)andP N(P) = [N(P) - N(P)]/N(P) (5.362)Then the nonanalog game is more efficient than the analog, i.e., E > E ifPo(P) + PN(PV[I - p N(P)] > O (5.363)Thus, if both the relative variance difference, p D, and the relative collision-number difference,p N, are positive, then the nonanalog game is clearly superior to the analog game. Themajority of the commonly used nonanalog games, however, will only decrease either thevariance or the number of flights, compared to the the analog game, but very seldom bothof them.Theorems 5.12 and 5.20 reflect clearly this antagonism since whenever the sufficientconditions of variance reduction (formulated in Theorem 5.20) hold, the condition of Theorem5.12 (that would ensure a decrease in computing effort) is likely to fail and vice versa.The heuristic explanation of this contradiction is that nonanalog kernels that satisfy theconditions of Theorem 5.20 force the particles toward regions from where contributions tothe score are expected, i.e., they make the particles suffer a higher number of collisions inthe important regions.Finally, it is also noteworthy that variance reduction techniques often require highercomputing time per collision (or per free flights) than the analog game as a consequence ofthe more complicated kernels to be sampled. Therefore, any technique that is supposed toincrease efficiency should be examined very carefully before it is applied routinely.J. OPTIMIZATION OF SOURCE DISTRIBUTIONIn this Chapter, we have investigated the possibility of variance reduction by suitablealteration of the kernels that govern the simulation (nonanalog games) and by proper choiceof contribution functions (estimators). A third possibility of influencing the total varianceof a Monte Carlo game is the use of appropriately chosen nonanalog source densities.Alteration of the particles' source density changes the starting weights of the particles, buthas no effect on the simulation of the migration, i.e., source biasing does not necessitateessential changes in an analog simulation scheme.Let us consider a problem with an analog source density Q(P). Let M 2(P) be the secondmoment of the score due to a particle starting from P with a weight of unity. Assume thatthe source density is altered to some Q(P). Then the weight of a starter at P in the alteredgame follows from Theorem 5.9 asW 1,- Q(PVQ(P)