Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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262 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsnormalized to unity and their difference may have no constant sign. On the other hand, thetheorem implies that a distortion of the kernels, which increases their values in importantregions (where the contribution and the expected score is large) and decreases them at lessimportant points, will probably yield variance reduction.The theorem gives conditions of variance reduction in terms of the kernels that describea transition from a starting point (or a postcollision point) to the next postcollision point.For symmetry reasons, it is expected that similar conditions can be constructed in terms ofthe transition probabilities between two successive precollision points. This is indeed so,and it is demonstrated by the example of a nonmultiplying game in which only free flightsbetween two collisions contribute to the score, i.e.,f a(P') - f s(P',P") = 0Let3~(P,P') =T(P,P')w(P,P')andr€(P',P") =c s(P')C s(P',P")w s(P',P")Then we writeIf(P,P") -JdP'9"(P,P') c €(P',P")Furthermore, let us introduce the notationsm,(P') = JdFC(POR)M 1(P") i = 1,2andm,(P") = JdF^(POF)M 2(P")Then the difference of the analog and nonanalog second moments follows from Equations(5.295) through (5.299) asA(P) = JdP'T(P,P0[l w(P,P0][f 2 (P,P0 + 2f(P,P0m,(P0 + m 2(P0][dP'37P,P0[m 2(P0- ITi 2(PO]It is seen that A(P) > 0 if w(P,P0 < 1 and m 2(P0 > rh 2(P0. On the other hand, it followsfrom Equations (5.295) and (5.297) thatrh 2(P) = JdP'W.POjdFgdPOP")|f 2 (P\P") + 21XPOPOm 1(F) 4- m 2(P")] (5.313)and similarly for the analog momentm 2(P) • dP'C(P,P0 dP"T(P\P") [P(POF) 4- 2IfPOFOm 1(F) + m 2(F)] (5.314)
Subtraction of Equation (5.313) from (5.314) shows that m,(P) > Ih 2(P) ifC(P,P')T(P',P") > ^P 1PO SOPOP")and we have the following.Theorem 5.21 — A partially unbiased nonmultiplyit j .k •U •) ,. ^ < > .• * •are assigned to intercollision flights only has a variance NOR . i1I > >•, n • h. >,>h JO.analog game with the same contribution function if fo> ft. < ( 0C(P,P')T(P',P") ss C(P,P')T(P',P") (5.316)for every P, P', and P" such that f(P', P") > 0 or m 2(P") > 0.Generalization of the theorem to multiplying games goes along the lines t A' # . 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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Page 317 and 318: 305Chapter 6SPECIAL GAMESIn the maj
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Page 321 and 322: 309Equation (6.1) will be referred
Page 323 and 324: 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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