Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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214 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsLet M 1(P) denote the expected value of the computing time necessary for the simulation ofa nonanalog history that is started from the point P. We shall suppose that the time of playingany event is independent of the coordinates of the event. In the majority of the computerapplications, this assumption is justified. Furthermore, we assume that playing an absorptiondoes not necessitate extra computing time and the times for selecting from the various kernelsin a collision (scattering, n-fold multiplication) are equal to each other.In view of the assumptions above, computing time is spent for two types of events; atime if is necessary for playing a free flight (including the check of a possible escape andthe selection of the type of collision) and a time t cis spent by the selection of the postcollisioncoordinates. Obviously, these time values are independent of the weight of the particle.Note that these times can be considered as contributions of the respective events in anonanalog game, while the weight of the particle is kept unchanged. Summing up thesecontributions, an estimate of the computing time per history is obtained and the momentequations can be used for the analysis of the estimate. More rigorously, M 1(P) satisfiesEquations (5.129) through (5.131) with the contribution functionsf(P.P') = t f, I) 1(P') -- 0f,(P',P") = f„(P',P") =t cand with the weightsW' = W" = W a = W^' = WHenceM 1(P) = t f+t c|dP'f(P,P')c(P')+ JdP'f(P.P')|dP"C(P',P")M t(P") (5.190)This Equation may play a role in the a priori estimation of the efficiency of various games.In most cases, it is satisfactory to work with a simpler form of the equation where weassume that the computing time per history is simply proportional to the number of flightsto be simulatedM 1(P) = I 0N(P) (5.191)where N(P) is the expected number of flights in a history started at P. This is the case ifthe time of playing a flight is t f= t„, while t c= 0. Obviously, N(P) is the solution of theequationN(P) = 1 + |dP'f(P,P')JdP"C(P',P")N(P")= 1 •f JdF'L(P,P")N(F) (5.192)On the basis of Equations (5.190) and (5.192), it is easy to establish conditions underwhich a nonanalog game requires less computing time than a corresponding analog game.For the sake of simplicity, let us consider the number of flights as a true measure of thecomputing effort and consider the analog equation corresponding to Equation (5.192):N(P) -- 1 + dP"L(P,P")N(P")
2HLetn(P) = N(P) -N(P)Then n(P) satisfies the equationn(P) = fdP"[L(P,P") - L(P,P")|N(P") + jdF'L(P.P")n(P")Now if the nonanalog game is feasible, then this equation ha i \ w ot» -••." •to Theorem 5.2, n(P) > 0 if L(P,P") > L(P,P") at every , i , .Thus, we have the following.Theorem 5.12 — A nonanalog game with the kernea lower expected number of collisions (flights) than the ansatisfy the inequality\dP'T(P,P')C(P'.P") >(dP'T(P,P')C(P\P")for every P,P" such that N(P") > 0.Note that this simple condition is sufficient but not c < • . < • i imay be needlessly strict. Nevertheless, in many practice' • i.,i< , i
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Library of Congress Cataloging-in-P
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III. Statistical Considerations •
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IV. Further Generalizations — 183
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Appendix 6A:Unbiased Estimation of
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2 Monte Carlo Particle Transport Me
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Chapter 2INTRODUCTIONWhen we starte
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thus,As it is proved" — and is so
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9B. THE PROBABILITY MIXING METHODTh
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11It is easy to prove 15by summing
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13but the most probable values:x, =
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ISand give a random sign (by the us
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18 Monte Carlo Particle Transport M
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100 Monte Carlo Panicle Transport M
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102 Monte Carlo Particle Transport
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305Chapter 6SPECIAL GAMESIn the maj
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3©7A. CORRELATED MOMENT EQUATIONSI
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309Equation (6.1) will be referred
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311The second moment of the score d
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313andC S(P',P")/C S(P',P") = y(P',
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315introduced in Equation (6.1) for
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317andL„(P 0,P;,...,Pi + 1) = L n
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319whereB„(P„,P; P:, ,) = A*(P
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E. EXAMPLES AND SPECIAL TECHNIQUESL
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323The weight factors associated wi
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325It is seen from Equation (6.37)
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327G. PARAMETRIC PERTURBATIONS: INT
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129Hall 31 gave a constructive deri
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331kernels, which remain unchanged
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333and from Equation (6.51)w';,,(i)
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335According to Equations (6,55) th
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337Minimization is performed by set
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339and letdsdaLet us consider a cor
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341whereis the length of the flight
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and from Equation (6.79)I / ds \ 2
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345Let us assume that we are intere
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347Introduction of this factor also
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3492. Let i|/ n (P) denote the solu
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3SJNow, if ( "> > 0, then k = i an
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353andk„ = S l "VS'" •" (6. if;
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3SSLet k„ denote the normalizatio
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3572. It is then proved that with t
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and start new histories until N new
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361The application of source iterat
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363we haveD 2 [k] = < 2 (k, - k,) )
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365neutron density, S 0 for every
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367if the same relation holds for t
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369Then the multiplication factors
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371in the unperturbed system, the w
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373The perturbation of the effectiv
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375where P" = (r',E'). Multiplying
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377iterate of the derivative of the
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579where P* = (r*,w*,E) = (r*,E*).
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381This estimator, unlike f, in Equ
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383tends to some stable attracting
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3§SIo the case of the next-event e
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387LDFIGURE 6.1.Geometry in scoring
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389Let us denoteThen Equation (6.18
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391is to be increased in order to o
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393Now, since bl(P) = g(P) is bound
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395is scored at every collision wit
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397and its expectation isR — R 1n
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3994. If O > 0 m, then the particle
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401plays a distinguished role), and
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403Theorem 6.7 — If x and s denot
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this a priori information, the best
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40?Thus, the bias introduced into t
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409In view of Equation (6.226) and
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41.1This means that if we use the k
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413Then the whole-sample and rare-s
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41.5According to the results of Sec
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41?(Note that vD 2 [|i|v] is indepe
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419withkOn the other hand, § is an
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421Thus, the expected ratio reads
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423St follows from Equations (6.265
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425Wc start from the observation th
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427Nowi'6.77K)kj„,and therefore(V
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42*3Similarly, for the fourth momen
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43 sOn the other hand, multiplying
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43.¾This will be proven in the lem
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435ThenT.m = EJ.Iy.y.Rjeyiy,,,and s
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6. Brown. F. B. and Martin, W. R.,
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67. Rief, H. and Fioretti, A., Mont
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Q = iy.M 2 2 (1/T, - 1/1,..,)/1, £
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48© Monte Carlo Particle Transport
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a ( X506 Monte Carlo Particle Trans
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513INDEXAAbsorptiondefined, 25photo
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515score probability in general tim
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517path stretching, 447—455splitt
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Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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