Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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274 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsi.e.,X(P) =M 1(P)Henceg(P,P') = M 1(P) - M 1(P') (5.340)i.e., the optimum composed estimator is the difference of the expected scores at the startingand end point of a flight. 26Variations of Equation (.5.337) with respect to the other contributionfunctions provide the equation systemf(P,P') + f,(P') -M 1(P)f(P,P') + f s(P',P") = M 1(P) - M 1(P")andf(P,P') + f rl(P',P") + (n - I)JdP"C n(P',P")[f„(P',P") + M 1(P")] - M 1(P) - M 1(P")It is easily seen that the solution of the equation system that also satisfies Equation (5.340)isf(P,P') = M 1(P), f a(P') = 0 (5.341)andf s(P',P") = f„(P',P") = ~M,(P") (5.342)Equations (5.341) and (5.342) define the minimum-variance partially unbiased estimatorsand the following theorem holds.Theorem 5.23 — The minimum-variance partially unbiased estimator set in Equations(5.341) and (5.342) yields a zero-variance estimate.Proof. Substituting the estimators into Equations (5.319) through (5.321) of the secondmoment, it can be seen that M 2(P) = M 1(P). Instead of the formal proof, however, let usrealize that the contribution functions in Equations (5.341) and (5.342) score M 1(P 1)M 1(P 1 +,) in the i-th flight (that starts from the collision point P 1) if it is followed by a realcollision and M 1(P,) if it is the last flight (followed by an absorption). Therefore, the finalscore in a history started from P is always M 1(P), with no fluctuation.Note that we have also shown in passing that the minimum variance-composed estimatorhas the form in Equation (5.340). Naturally, this estimator is no more feasible than the zerovarianceestimators above. Nevertheless, it will be seen in Section H that there exists apartially unbiased estimator which approximates rather well this optimum. It can be seenfrom Equations (5.319) through (5.321) and (5.3.39) that in a game where the optimumcomposed estimator g(P,P') scores in the intercollision flights and zero contributions followfrom the collisions, the variance of the score satisfies the equationD 2 (P) = JdP'T(P,P')V(P') + JdP'T(P,P')JdP"C(P',P")D 2 (P") (5.343)
275withV (P') =dP" C(P',F) Mf(F)dP"C(P', F)M 1(P")+ Ct(P') 2 "(n ~ Oq n(P')n - 1dP" C 11(P' ,FJM 1(P") (5 344"!Another interesting consequence ot the derivation above is that although the composes!estimator is the expectation of the reaction-dependent contributions over the possible outcomesof the collision and, as such, one might expect a lower variance, it does not necessarilydecrease the variance of the score, compared to the reaction-dependent estimators. Nevertheless,in most practical cases, the estimators do not depend on the type of collision (cf.Section 5. VI.B) and therefore approximations to the minimum variance-composed estimatorhave lower variance than the usual estimators.In the following section, we compare the variances of the commonly used estimatorsin an analog simulation. A corresponding analysis for nonanalog game may be performedin a similar way.G. RELATIVE MERITS OF THE COMMON ESTIMATORSWe consider here the estimators derived in Section 5.VLB. Let us first notice thai(except for the last-event estimator) all these estimators are the composed type, i.e., theydo not depend on the type of reaction at the end of the free flight and they only depend onthe starting point, P, and end point, P', of the flight. Comparison of the variances of reactiondependentestimators in nonmultiplying games is reported In References 28 and 31.The second moment of the score in an analog game with the estimator f(P,P') followsfrom Equations (5.319) through (5.321) asM 2(P) = dP'T(P.P') f 2 (P,P') + 2f(P,P') dP"C(P',P")M,(P")+ dP' T(P,P') dP"C(P' ,P") M 2(P")(5.345)We shall follow the usual procedure of variance comparison, i.e., we examine the differerw;;.-of the source terms of Equation (5.345) for estimator pairs. If A(P) denotes the differenceof the variances with the estimators f,(P,P') and f 2(P,P'), thenA(P) = A(P) +dP"L(P,P")A(P")whereA(P) = |dP'T(P,P'){f 2 (P,P') - {l(P.P')2[f,(P,P') - f 2(P,P')]LDP"C(P',P")M4P")} (5.34?;The variance of the score by the estimator f, is greater than that by C 2if A(P) > 0 for everypoint P of the domain of simulation (sufficient condition).Let us first compare the variance of the score with an arbitrary estimator of the formf,(P,P') =f(P,P')
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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 236 and 237: 224 Monte Carlo Particle Transport
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Page 317 and 318: 305Chapter 6SPECIAL GAMESIn the maj
Page 319 and 320: 3©7A. CORRELATED MOMENT EQUATIONSI
Page 321 and 322: 309Equation (6.1) will be referred
Page 323 and 324: 311The second moment of the score d
Page 325 and 326: 313andC S(P',P")/C S(P',P") = y(P',
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Page 331 and 332: 319whereB„(P„,P; P:, ,) = A*(P
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Page 335 and 336: 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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