Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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92 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsThe separation and analytical integration of a part of the function to be integrated isonly one way of using expected values. The reader has already met similar examples: theuse of the statistical weight discussed in Section 3.II.C is a replacement of absorbtion byits probability, and Section 3.II.D. was fully devoted to the use of expected values in scoringduring random walk simulations.F. CORRELATED SAMPLINGCorrelated sampling is a very effective variance reducing technique when one wants tocalculate the effect of a small perturbation in the system analyzed, or to study several similarproblems, or — in our present approach — to compute the difference between two integrals.In such cases the statistical uncertainties of two (or more) independent straightforward MonteCarlo calculations may happen to be in the same range as the small difference to be calculated.In other words, the result may be lost in the random noise. However, if the simulations usethe same sequence of random numbers, the difference in the results will more clearly exhibitthe real difference between the problems.As an illustration of this technique, let us follow the example of McGrath and Irving. 20The task is the calculation of the difference between two integrals:AI = I, - I 2where1 1= JdXh 1(X)Cp 1(X)and1 2== Jdxh 2(x) (p 2(x)In a straightforward simulation twice N independent samples X 11, x, 2, . . . ,x, N; X 2 1,x 22, . . . ,x 2Nare picked up from the PDFs (p, and
93If the estimations of h, and h 2are positively correlated, thencovin,,h 2) > 0and the resultant variance is less than that resulting from the use of two in.. •. «forward samplings. Application of correlated sampling to particle trans;» >> 1detailed in Chapter 6.1.G. FURTHER METHODSThere are many more methods to decrease the variance »i \ ,N «- C , i >,listed here. Several techniques with special importance in netfamiliar to the readers of Chapter 3, other procedures will h ' , < tof the book.For a more complete summary one of the basic references' 4 ' 2 ' 1 ' 2 " should be consultedIII. SOLUTION OF FREDHOLM-TYPE INTEGRAL EQUATIONSA. INTRODUCTIONIn this Chapter, the simple integral solutions discussed above will bstep to arrive to Fredholm-type integral equations of the second kind. •of the particle transport equations.Let us denote the PDF in Equation (4.5) by tp, and assume ,> , ,. i «V'»»equation, i.e., let us write• 'I = jdxh(x)(p,(x) (4..24]Furthermore let us assume the cp, is expressed asalso a PDF.Let us introduce the conditional probability k as:, , K(x',x)uxx') = -y^w(x )wherew(x') == JdxK(x',x)process:Theorem 4.7 — The integral I of Equation (4.24) can be estimated1. Choose N values of x[ from
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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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142 Monte Carlo Particle Transport
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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
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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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484 Monte Carlo Panicle Transport M
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486 Monte Carlo Panicle Transport M
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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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