Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

285Since no other biasing is assumed in the altered game, the second moment of the score dueto a starter from P isM 2(P) =Wf 1(P)M 2(P)and the total second moment of the score in the source-biased game isM 2= JdPQ(P)M 2(P) = JdPQ(P)Wf 1(P)M 2(P) = JdPQ 2 (P)M 2(P)/Q(P)provided the altered source is normalized to unity, i.e.,IdPQ(P) = 1The nonanalog source density is to be chosen such that the total second moment M 2beminimum. This means that the optimum source density is the solution of the variationalproblemJdP[Q 2 (P)M 2(P)/Q(P) - XQ(P)] = minwhere X is the Lagrange multiplier due to the normalization condition. Solution of theextremum problem leads to the following theorem.Theorem 5.26 — The minimum variance source density of an otherwise analog MonteCarlo game has the formQ(P) = Q(P) VM 2(P)/j dP'Q(P') VM 2(P')The variance of the game with this density isD- = JdPQ(P)VM 2(P)J 2 - JdPQ(P)M 1(P)LJPractical realization of the optimum source density would require a priori knowledge of thesecond moment M 2(P) of the analog score at every point P. With point sources, this momentcan be estimated at a relatively low extra cost, but for extended sources, determination ofthe pointwise second moment is not possible. Hoffman 15proposes a version of the abovetechnique in which the probabilities of starting a particle from various phase-space regionsare altered in an optimum way. Let the entire phase space over which the source is extendedbe divided into N distinct regions denoted by (1), (2), (N). Let Q nbe the probabilitythat a particle starts from the n-th region in the analog game. Then obviouslyQ n=J dPQ(P)Let the altered game be such that the particles start from region n with a probability Q n,but let their distribution inside the region be the same as in the analog game. Accordingly,the altered source density readsQ(P) = Q n[Q(P)ZQ 1J if P e (n)