Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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230 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswith some D 1that satisfy2 b,(P,P,) = i (5.218)1 = UCombining the results of the theorems above, it is seen that given a partially unbiasedestimator set S{G,JGJ^ 0}, a new set of estimators S{G,{G,}^. 0} can be generated by transformingg (the score in a flight) according to Equation (5.214) and the reaction-dependentscores gj with the aid of Equation (5.216).For the sake of completeness, we show that any partially unbiased estimator is thetransform of the simplest estimator f(P').Theorem 5.17 — If 8{0,IgJfL 0} is a set of partially unbiased estimators, then thereexists a transformation of the form in Equation (5.216) that brings the basic estimatorSIO.F.F.IFJ^JINTOSIO.IG^J-Proof. It is to be shown that there exist functions X 1(P 5P 1,?') and ^(P 5P 1) (i = 0,1,...) suchthat&(P,P') =/dP^KP^OXXP^^PVB^P,?,)^,)/JdP 2T(P 5P 2)X 1(P 5P 11P 2)P 1(P 2) J (5.219)where the b ;'s are related to the quantities in Equation (5.215) asb,(P.P.) = 2J=Oa.,(P,P,)and2 b,(P,P,) = 1i=0Simple substituions shows that if the g2s form a set of unbiased estimators, then they followfrom transformation (5.219) by puttingX,(P,P,.P') - gi(P,P')andb,(P,P,) = dP'T(P 5P') g l(P,P') P i(P')/ dP'T(P,P')f(P')•Finally, a trivial feature of the partially unbiased estimators is stated.
Theorem 5.18 — Any linear combination ofpartially unbiased, provided the Smear combinatu i> « .S{G' K '{{G 1' K, }R,.„} (K - 1,2 N) are sets of partialb -nb-tIn the following Section, the commonly used estimators are introducedformation of the weighting function f(P).B. COMMONLY USED ESTIMATORSThe simplest unbiased estimator was seen to be the weighting function f(Prate (5.2). The estimator is called the collision estimator and it scores f(Pjpoints Pj. If one is only interested in an unbiased estimate of the reaction r a < \,of the computational effort involved, then this estimator serves the purposeis no reason to seek other possible contribution functions. In practical case'amount of computation necessary to reach a given accuracy is almost as im.x<
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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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