Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

391is to be increased in order to obtain a more reliable estimate in a given amount of computingtime.Second, let us observe that sampling a distance D 1(P) in Equation (6,184) is essentiallyequivalent to the method of "pseudo scattering" or "delta scattering" (Section 5.V.H).This means that the estimation procedure above can also be introduced with the aid of anonanalog transition kernel that defines delta-scattering events with a certain probability. Amodification of the next-event estimator with delta-scattering events was proposed byMikhalov 58and was shown to exhibit a 1/| r — r*| singularity. Although this type ofsingularity in an analog game would result in a bounded-variance final score, introductionof delta scattering (as a nonanalog event) increases the frequency of collisions in the neighborhoodof r*, and therefore the distance r = |r — r*| between the collision points and thedetector point is distributed approximately like 2rdr instead of the analog distribution 3r 2 dr.Consequently, this type of singularity also results in an unbounded variance' 18and a convergencerate of Vlog en/n.I). BOUNDED-VARIANCE POINT ESTIMATORSSince Theorem 6.6 excludes the possibility of finding a bounded-variance partiallyunbiased estimator, any estimator with bounded variance must account for not only an.intercollision free flight, but also the collision process preceding it. Let us consider a partiallyunbiased estimator with a 1/b singularity, as given in Equation (6.185), i.e., letwhere g(P,P') is bounded and b is the distance of r* from the flight direction. Assume thata particle with a direction to, and energy E 0enters a collision at r. The expected score inthe next free flight(s) of the progeny coming out of this collision is(6,186)where P 0= (r,W 0,E 0) and I(P) is the expected partial score in the next flight from P : =(r.to.E) of a particle. According to Equation (6.182), it has the form1I(P) = - g(P) (6.187)where g(P) is bounded. Equation (6.186) is rewritten aswhere againand(6.189)C 2(w 0-^w|r,E,E 0) -- C(w 0,E 0-^to,E|r)/C 1(E 0-^E|r,w 0) (6.190)