Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

494 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsrealizations may be different. Therefore, it may be useful to introduce zero-variance pathstretchingschemes with estimators different from the last-event estimator. In what follows,we introduce a scheme with the expectation estimator.The contribution of a flight from a point P is then equal to the expected first flight score,I 1(P), in Equation (7.135), i.e.,f(P,P') =I 1(P)A zero-variance scheme follows directly from the general form derived in Section 5.VlITA.Inserting the expectation estimator above into the general Equation (5.288), the transformedtransition kernel readsT(P 1P') - T(P,P')[Ij(P) + JdP"C(P',P")M 1(P")]/M 1(P) (7.153)Introducing again the adjoint collision density, two equivalent forms of the transition kernelfollow from Equations (7.149), (7.150), and (7.152) asf(P,P') - T(P,P')[I,(P) + ^(POJ/MXP) - T ( p ' p ') (7.154)The transformed collision kernel in a general zero-variance scheme is given in Equation(5.290), and it is seen that the collision kernel is independent of the contribution function.Thus, in the zero-variance leakage estimation, this kernel is the same as the one given inEquation (7.148) or (7.151) for both the last-event and expectation estimators.There is one point to be emphasized here. The scheme with the expectation estimatoras derived above is not an exponential transformed game. This is seen from Equations(7.153) and (7.154) since the terms multiplying the analog transition kernel on the RHS ofthe equations do not factorize to functions depending separately on P and P', respectively,and therefore T(P,P') in these equations does not conform to the exponential transformed(or importance-sampling) form of Equation (7.137). Therefore, realization of the scheme(if it were possible at all) would only be practicable by the use of a nonanalog simulationand statistical weights. Nevertheless, approximately optimum path stretching can also bedefined with an expectation estimator, as will be discussed in the next section.B. DISCUSSION OI THE SCHEMESBefore turning to practical realizations, we shall briefly discuss some specific propertiesof the schemes above.Let us first note that the probability of an absorption is zero in both schemes, i.e., thetransformed collision kernel is normalized to unity:JdFC(P',F') - 1This is a common property of all partially unbiased zero-variance schemes, as pointed outin Section 5. VIII.A. Unit survival probability is reached by transformation of the collisionkernel. This particular property of a nonanalog game can also be produced by survivalbiasing (cf. Section 5.VIII.D). In this case, the nonanalog collision kernel isC(P',F) = C(P',F) /dP"C(P',P")