Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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422 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsand(6.267)Proof.haveTaking the expectation of f and repeating the derivation of Equation (6.260), we(r) = — (m,y - m 2m,x — m,m,/x — m,\2[1 + 5,7(Xy) + s,,/x 2 ])O 1^ (1 - (y ^ m2)(* - m,) /x - m+ + O 3m, m 7m, V m, m 7m, mtm 2m,V 1 2- (S 1 2) V? - (S 1 1)m,m.O 3It remains to show that the terms in square brackets reduce to unity. Let us calculate theexpectation of the empirical covariance in Equation (6.266):n — I n 1H i , n - 1 n , n ini1- V,whereV 1 2= the covariance of the i-th realizations. (Naturally, it is independent of i since the realizationsare identically distributed.) On the other hand, the covariance of the sample means isV ,2= (A S x, 2 y.) -MIII 2 = (4 E x,y, + ~~ 22 x,y,>n j j Ii i iin(n - 1)m,m 9---- - V,i.e.(S 1 2) = V 1Similarly, it can be seen that
423St follows from Equations (6.265) through (6.267) that the correction to y/x declike 1/n, i.e., for a sufficiently large number of averaged realizations in x and y, thin the estimate f becomes negligible.Having obtained an unbiased estimate of the ratio, it is natural to ahof its variance. We shall first determine the theoretical variance of f, thiof it in terms of the sample statistics. Since the density function of x and \ , , ibe concentrated in a small region around the expectations, terms of the order'AI 1 "•)(k & OSare small compared to the second-order quantities and therefore terms proportional to fourthand higher relative central moments will be neglected.Let us first notice that the variance of the corrected empirical estimate f in Equator(6.265) differs negligibly from that of f in Equation (6.264). IndeedD 2 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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25!) 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
Page 383 and 384: 371in the unperturbed system, the w
Page 385 and 386: 373The perturbation of the effectiv
Page 387 and 388: 375where P" = (r',E'). Multiplying
Page 389 and 390: 377iterate of the derivative of the
Page 391 and 392: 579where P* = (r*,w*,E) = (r*,E*).
Page 393 and 394: 381This estimator, unlike f, in Equ
Page 395 and 396: 383tends to some stable attracting
Page 397 and 398: 3§SIo the case of the next-event e
Page 399 and 400: 387LDFIGURE 6.1.Geometry in scoring
Page 401 and 402: 389Let us denoteThen Equation (6.18
Page 403 and 404: 391is to be increased in order to o
Page 405 and 406: 393Now, since bl(P) = g(P) is bound
Page 407 and 408: 395is scored at every collision wit
Page 409 and 410: 397and its expectation isR — R 1n
Page 411 and 412: 3994. If O > 0 m, then the particle
Page 413 and 414: 401plays a distinguished role), and
Page 415 and 416: 403Theorem 6.7 — If x and s denot
Page 417 and 418: this a priori information, the best
Page 419 and 420: 40?Thus, the bias introduced into t
Page 421 and 422: 409In view of Equation (6.226) and
Page 423 and 424: 41.1This means that if we use the k
Page 425 and 426: 413Then the whole-sample and rare-s
Page 427 and 428: 41.5According to the results of Sec
Page 429 and 430: 41?(Note that vD 2 [|i|v] is indepe
Page 431 and 432: 419withkOn the other hand, § is an
Page 433: 421Thus, the expected ratio reads
Page 437 and 438: 425Wc start from the observation th
Page 439 and 440: 427Nowi'6.77K)kj„,and therefore(V
Page 441 and 442: 42*3Similarly, for the fourth momen
Page 443 and 444: 43 sOn the other hand, multiplying
Page 445 and 446: 43.¾This will be proven in the lem
Page 447 and 448: 435ThenT.m = EJ.Iy.y.Rjeyiy,,,and s
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Page 451: 67. Rief, H. and Fioretti, A., Mont
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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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