Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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338 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsparameter values. Taking full advantage of this assumption, the Taylor series of the reactionrates around a has the formr(a) = i-(«) + d • (a - a)and (6.63) reduces to3 1 R ~ 1 Jr(Ot) - m + U • (a - r)] + A "' • (a - a) = O (6.66)After rearrangement, the difference of the measured and calculated rates readsm - r(a) = [Rfli' 1 ' 'A + dj(a - «)Taking into account the identity[RtT"'A •' = AS 1 jR + SAAS T 1the adjusted parameters are expressed asa tx •- Aa 1 JR + aa]-'[m - r(a)] (6.67)In this expression, the covariance and sensitivity matrices are taken at the original valuesa of the parameters. Similarly, the covariance matrix of the adjusted data follows fromEquation (6.65) asA = H~'(a) A Va[R I dAd ]" l dA (6.68)Equations (6.67) and (6.68) were derived by Dejonghe et al. 8 driectly from maximumlikelihood principles. This approach seems to have the advantage that no iteration is necessaryfor the adjustment, i.e., the reaction rates and related matrices have to be estimated onlyonce. Note, however, that the two approaches are not equivalent in spite of the fact thatboth are based on the approximation that at certain points the derivatives of R and d canbe neglected.In fact, Equations (6.63) and (6.64) are also valid if d depends on the parameters andits derivative is neglected solely in the expression of H in Equation (6.65). Thus, the IteratesH n+, are evaluated with updated values of d(a,). In contrast, Equation (6.66) assumes aconstant d . It can be easily seen that the adjusted parameter set given by Equation (6.67)is just the first iterate a,, as follows from Equations (6.63) through (6.65). Hence, weconclude that the second approach defines the first-order adjustment of the parameters, andfurther iterates by the first approach give adjustments depending on higher-order derivativesof the reaction rates with respect to the parameters.I). ESTIMATION OF HIGHER-ORDER DERIVATIVESHigher-order derivatives of the reaction rate in the expansion (6.45) can also be estimatedby Monte Carlo. The weight generation rules of higher-order differential games follow fromrepetition of the line of thought of the previous and present Chapters. Let us denoteS , , )_i!idaa
339and letdsdaLet us consider a correlated game estimating a function of s (i, and s (i)in which diparticles are associated with the systems of parameter values a and a 4 \
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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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102 Monte Carlo Particle Transport
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25!) Monte Carlo Particle Transport
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Page 300 and 301: 288 Monte Carlo Particle Transport
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',
Page 327 and 328: 315introduced in Equation (6.1) for
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)
Page 339 and 340: 327G. PARAMETRIC PERTURBATIONS: INT
Page 341 and 342: 129Hall 31 gave a constructive deri
Page 343 and 344: 331kernels, which remain unchanged
Page 345 and 346: 333and from Equation (6.51)w';,,(i)
Page 347 and 348: 335According to Equations (6,55) th
Page 349: 337Minimization is performed by set
Page 353 and 354: 341whereis the length of the flight
Page 355 and 356: and from Equation (6.79)I / ds \ 2
Page 357 and 358: 345Let us assume that we are intere
Page 359 and 360: 347Introduction of this factor also
Page 361 and 362: 3492. Let i|/ n (P) denote the solu
Page 363 and 364: 3SJNow, if ( "> > 0, then k = i an
Page 365 and 366: 353andk„ = S l "VS'" •" (6. if;
Page 367 and 368: 3SSLet k„ denote the normalizatio
Page 369 and 370: 3572. It is then proved that with t
Page 371 and 372: and start new histories until N new
Page 373 and 374: 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
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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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Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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