Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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276 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsand the variance with the expectation estimator corresponding to itf,(P,P') = f E(P) =jdP'T(P,P')f(P,P')The difference of the source terms in this case readsA(P) = dP'T(P,P')f 2 (P,P') dP'T(P,P')f(P,P')2 dP'T(P,P')lf(P,P') - f E(P)]M,(P')(5.347)orA(P) = dP'T(P,P'){[f(P,P') + M 1(P')] 2 - [f E(P) + M 1(P')]'}withM 1(P') =JdP"C(P',P")M,(P")The difference of the first two terms in Equation (5.347) is certainly nonnegative and thereforethe following theorem holds.Theorem 5.24 — A partially unbiased estimator yields greater variance than the correspondingexpectation estimator iff E(P)
277score by the expectation estimator is definitely greater than that of certain other partiallyunbiased estimators. 28 - 42For example, for a not-too-large probability of absorption (not-too-small number ofsecondaries per collision), the track-length estimator has, indeed, a lower variance than theexpectation estimator. 28On the other hand, it is also seen from Equation (5.347) that with increasing probabilityof absorption, the third term in the expression of A(P) decreases in modulus, i.e., forsufficiently strong absorption, the positive difference of the first two terms is dominant andthe expectation estimator becomes more efficient than any other partially unbiased estimator.In the limiting case of a purely absorbing medium, the expectation estimator will be identicalto the optimum composed estimator in Equation (5.340), i.e.,I) n(P) = M 1(P) = jdP'T(P,P')f(P,P') (5.348)with zero variance.Although in most practical cases the expectation estimator provides the smallest varianceof all the partially unbiased estimators, it should be taken into account that when using it,one has to evaluate an exponential function at every collision. Since this procedure is rathertime-consuming, the track-length and collision estimators may find their application evenin such problems where their variances are expected to be somewhat higher than that of theexpectation estimator.Next, we consider the variances of the collision and track-length estimators in collisionrateestimations. Let us assume again that the collision rate is to be estimated in a barebody, i.e., no particle enters the body from outside. In this case,f,(P,P') = 1f 2(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 238 and 239: 226 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 329 and 330: 317andL„(P 0,P;,...,Pi + 1) = L n
Page 331 and 332: 319whereB„(P„,P; P:, ,) = A*(P
Page 333 and 334: E. EXAMPLES AND SPECIAL TECHNIQUESL
Page 335 and 336: 323The weight factors associated wi
Page 337 and 338: 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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