Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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414 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsand|m| 5* |{m*>|•It can also be seen 49thatlim (r) = +oc (6.243)andlim p(r) = k/n (6.244)P->-0Accordinglylim m* = 0p-->0i.e., for very rare events (p 1), the Gaussian estimate m* predicts a practically zero valueof the nonzero common mean m. This bias is often aggravated by human error in thatoccasional and extremely high contributions among a great number of low ones (zeros inour case) are intentionally discarded as "random fluctuations". Both Theorem 6.10 and thepossibility of wrong subjective decisions suggest that whenever the occurrence of rare eventsis indicated, the obtained data should be checked thoroughly from the point of view of raritybefore statistical evaluation.In the case of independent rare sets, the number of realizations n ;as well as the numberof rare events V 1may be different for different random variables z'. Although in full generalitythe ''rarity probability" p may also be different for different variables, generalization of theconsiderations to such a case is troublesome. On the other hand, the assumption of a common"rarity probability" p is fairly realistic since in most practical cases the very same phenomenonis simulated independently by different methods and the different estimates (realizations)are combined.Let z U ) , z (2) ,...,z
41.5According to the results of Section A an unbiased estimate of the expectation of the rarecontributionsp. = m/pisk(1=2 (6,245.wherek£>" = (v,/s,_,.)/S (6.246)j =1andx (i) = X xf'/v,s, r= J (xf> - F>) 2 /(v, - 1)J= 1are the rare-sample statistics. It can be easily seen from the likelihood function of therealizations xj'> that the unbiased maximum likelihood estimate of the probability p iskki = ii = iTherefore, an unbiased estimate of the common mean in terms of the rare statistics readsm = P(I = (2 ",/¾ D 1) 2 cf/x® (6,,247)Again, this is a safe estimate if rarity is accounted for when generating the statistics. H,however, the whole-sample statisticsz (i! = 2zf ) / n i= v.x^Vn,ands, = 2 (zj" - z ,0 ) 2 (n, " Di=iare used, then the combination coefficients can be expressed asc'f- (r,n/s,)/X (lyi/Si)j=ik
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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
Page 375 and 376: 363we haveD 2 [k] = < 2 (k, - k,) )
Page 377 and 378: 365neutron density, S 0 for every
Page 379 and 380: 367if the same relation holds for t
Page 381 and 382: 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: 413Then the whole-sample and rare-s
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 and 434: 421Thus, the expected ratio reads
Page 435 and 436: 423St follows from Equations (6.265
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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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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