Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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94 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsProof. The expected value of 1 is:J.I-dxh(x) IP 1(X) =I•B. FREDHOLM TYPE INTEGRAL EQUATIONS, TUNC TIONALS TO BEDETERMINEDThe second-kind Fredholm-type integral equations are defined by:(4.26)where (p(x) is the multidimensional function to be determined, Q(x) is called the source termof the equation and K(x',x) is the non-negative integral kernel.If afunctional of the form of a weighted integral of
95whereI 1= JdXf(X)(Pj(X) (4.28)way:Theorem 4.8 — The integral I of Equation (4.27) can be estimated in the foil1. Set j = 0, select the initial coordinate x u, from the PDFt,q(x) =Q(x)W,,owherew M 1= Jdx Q(x)2. Select the (j + I)-St coordinate x M + 1from the conditional probabilityk(xx ld)wi,j( x ij)wherejdxK(x id;x)is called the (j + I)-St weight factor.3. Set j = j + 1 and return to step 2.4. After N repetition of the process the quantityI 4 I W 1Jf(X 1 0)is an unbiased estimate of Ij of Equation (4.28), if the Wy weight is defined as:w,j = ri W 1,,t -- 0A proof can be otained by recursive application of the proof of Theorem 4.7.The total functional I of Equation (4.27) is trivially the sum of the contributions of al!terms in the Neumann series:J = ODuring the execution of the procedure outlined in Theorem 4.8 steps 2 and 3 form aninfinite cycle which is to be stopped for any practical application by reasonable criteria.Cut-off criteria can be assigned to any accuracies if the terms I 1are monotonously decreasingfor increasing j-s.Special cut-off criteria will be given in the next Chapter — for the transport processes.
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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
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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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Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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