Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

25!) Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsquantitatively analyzed. It has been mentioned that there is little hope of finding exactsolutions to the second-moment equations in general, but very often we are only interestedin the relative merits of some procedures, i.e., we intend to decide whether a particulargame results in lower variance than another game. Usually, this can be done without effectivelysolving the equations. On the other hand, in certain applications, approximatecalculation of the variances (e.g., by the method proposed in the previous Chapter) givessufficient practical information.The theoretical variance of a game is connected to the score moments as follows. LetMj(P) be the i-th moment of the score due to a unit weight starter at the point P in thenonanalog game in question. Let Q(P) be the source density of the starters and letW(P) =Wbe the statistical weight of the starter from P. Obviously, the expected final estimate isdP Q(P)WM 1(P)and the variance of the estimate isD 2 [R] = JdPQ(P)W 2 M 2(P) -RIn this Section, zero-variance Monte Carlo schemes are first reviewed. Such schemesresult in final scores which have no statistical fluctuation, i.e., every history yields the verysame final score. Conditions are established in Section B under which the variance of afeasible game is bounded. Variance reduction capabilities of nonanalog games in generaland of special biasing schemes in particular are investigated in Sections C and D, respectively.The variance and efficiency of the equivalent nonmultiplying game are examined in SectionE. Minimum-variance partially unbiased estimators are derived in Section F. Comparisonof the variances of partially unbiased estimators is followed by the derivation of a new,effective, self-improving estimator. Some remarks concerning the effect of variance reductionon the efficiency of a game in general conclude the analysis of the second-moment equations.Finally, optimum biasing of the source density is addressed.A. ZERO-VARIANCE SCHEMESDiscussion of a Monte Carlo scheme that has no statistical error seems absurd sincesuch a game would give an exact answer in one history, i.e., in a quasideterministic way,while, in general, Monte Carlo is used only in cases where deterministic methods fail towork. It will be seen below that the suspicion concerning the practical feasibility of suchgames is justified indeed. Nevertheless, besides their theoretical interest, such games dohave practical importance in the sense that they represent the "best of all" Monte Carlogames that, in principle, can be arbitrarily closely approximated, and their structure indicatesthe directions of the approximations.Zero-variance schemes were first derived through a special importance sampling procedure,and these schemes involve last-event (absorption) estimators. 1218Zero-variancebiasing schemes with a collision estimator were introduced by Ermakov 10 and Hoogenboom. 14Schemes with arbitrary partially unbiased estimators were derived from the moment equationsby Dwivedi 9 and were generalized by Gupta. 13Both derivations concern nonmultiplying games. One might think that the form of theestimator is irrelevant if it is about a zero-variance game. This, however, is not so sincethe practically realizable games may only be approximations of the ideal one and will