Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

274 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsi.e.,X(P) =M 1(P)Henceg(P,P') = M 1(P) - M 1(P') (5.340)i.e., the optimum composed estimator is the difference of the expected scores at the startingand end point of a flight. 26Variations of Equation (.5.337) with respect to the other contributionfunctions provide the equation systemf(P,P') + f,(P') -M 1(P)f(P,P') + f s(P',P") = M 1(P) - M 1(P")andf(P,P') + f rl(P',P") + (n - I)JdP"C n(P',P")[f„(P',P") + M 1(P")] - M 1(P) - M 1(P")It is easily seen that the solution of the equation system that also satisfies Equation (5.340)isf(P,P') = M 1(P), f a(P') = 0 (5.341)andf s(P',P") = f„(P',P") = ~M,(P") (5.342)Equations (5.341) and (5.342) define the minimum-variance partially unbiased estimatorsand the following theorem holds.Theorem 5.23 — The minimum-variance partially unbiased estimator set in Equations(5.341) and (5.342) yields a zero-variance estimate.Proof. Substituting the estimators into Equations (5.319) through (5.321) of the secondmoment, it can be seen that M 2(P) = M 1(P). Instead of the formal proof, however, let usrealize that the contribution functions in Equations (5.341) and (5.342) score M 1(P 1)M 1(P 1 +,) in the i-th flight (that starts from the collision point P 1) if it is followed by a realcollision and M 1(P,) if it is the last flight (followed by an absorption). Therefore, the finalscore in a history started from P is always M 1(P), with no fluctuation.Note that we have also shown in passing that the minimum variance-composed estimatorhas the form in Equation (5.340). Naturally, this estimator is no more feasible than the zerovarianceestimators above. Nevertheless, it will be seen in Section H that there exists apartially unbiased estimator which approximates rather well this optimum. It can be seenfrom Equations (5.319) through (5.321) and (5.3.39) that in a game where the optimumcomposed estimator g(P,P') scores in the intercollision flights and zero contributions followfrom the collisions, the variance of the score satisfies the equationD 2 (P) = JdP'T(P,P')V(P') + JdP'T(P,P')JdP"C(P',P")D 2 (P") (5.343)