On-the-Fly Monte Carlo Dominance Ratio Calculation Using the Noise Propagation Matrix

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1 Progress in NUCLEAR SCIENCE ECHNOLOGY, Vol. 2, pp (2) ARICLE On-the-Fly Monte Carlo Dominance Ratio Calculation Using the Noise Propagation Matrix homas M. SUON,*, Paul K. ROMANO 2 Brian R. NEASE 3 Knolls Atomic Power Laboratory Bechtel Marine Propulsion Corporation, Schenectady, NY, USA 2 Massachusetts Institute o echnology, Cambridge, MA, USA 3 Bettis Atomic Power Laboratory Bechtel Marine Propulsion Corporation, Pittsburgh, PA, USA wo Monte-Carlo-based methods or computing the dominance ratio in reactor calculations are the Fission Matrix Method (FMM) the Coarse Mesh Projection Method (CMPM). he FMM allows an estimate o the dominance ratio to be computed beore the ission source has converged, but requires a ine mesh hence considerable computational resources or suicient accuracy. Conversely, the CMPM gives very accurate results on a coarse mesh with very little computational eort, but can be used only ater the ission source has converged. In this paper we describe a new method called the Noise Propagation Matrix Method (NPMM) that has the same coarse-mesh accuracy properties as the CMPM while also permitting an on-the-ly estimation o the dominance ratio during ission source convergence. Like the CMPM, the NPMM uses the noise propagation matrix (NPM) in determining the dominance ratio. he two methods dier, however, in how the matrix is used to obtain the dominance ratio. A new derivation o the equations used to compute the NPM in a Monte Carlo calculation is presented that eliminates an approximation made in an earlier work on the CMPM. It is shown that by using the improved expression or the NPM, the dominance ratio can be ound directly simply as the largest-modulus eigenvalue o the matrix thereby eliminating the more complicated time-series-analysis method used by the CMPM. Results or several problems are presented that demonstrate the validity o the method. KEYWORDS: Monte Carlo, dominance ratio I. Introduction he dominance ratio o a issioning system is deined as the irst harmonic eigenvalue divided by the undamental mode eigenvalue. Its signiicance lies in two distinct areas. First, it is a property o the physical system, hence is related to such aspects o the system as xenon stability. Second, it is a property o the transport equation, hence is related to such aspects o the solution method as the ission source convergence rate cycle-to-cycle correlation. Being able to determine the dominance ratio is thus important to reactor designers analysts as well as reactor physics methods developers. Accurate determination o the dominance ratio rom Monte Carlo calculations has been the subject o considerable study in recent years. 5) For many applications it is only necessary that the dominance ratio be an end result o the Monte Carlo calculation. here are some applications, however, in which an evolving estimate o the dominance ratio obtained on-the-ly during the calculation may be useul. It is known, or example, that using the stard power method or ission source iterations results in the tallies rom successive ission generations (cycles) being correlated, that this correlation must either be minimized or accounted or in order to determine accurate statistical uncertainties. 6) It is also known that the degree o correlation can be estimated *Corresponding author, homas.sutton@unnpp.gov c 2 Atomic Energy Society o Japan, All Rights Reserved. 749 rom the dominance ratio. One method o minimizing the correlation between successive tallies is by combining multiple successive ission cycles into batches, deining the tally variables over the batches rather than over the cycles. 7) Here, an estimate o the dominance ratio obtained during the inactive cycle phase o the calculation is helpul in automatically determining the number o cycles per batch to use during the active cycle phase. One means o estimating the dominance ratio using Monte Carlo is the ission matrix method (FMM). 8) he FMM involves the division o the coniguration space o the problem into a number o discrete cells. Element i, j o the ission matrix is deined as the expected number o direct progeny ission neutrons born in cell i due to a parent ission neutron born in cell j. An estimate o the ission matrix may be easily obtained in Monte Carlo criticality calculations, its eigenvalue spectrum determined by stard matrix algebra methods. he dominance ratio may then be estimated as the ratio o the irst-harmonic eigenvalue o the matrix to that o the undamental mode. While the undamental-mode eigenvalue o the ission matrix is a good estimate o the undamental-mode eigenvalue o the Monte Carlo calculation, the same is not generally true o the irst-harmonic eigenvalue thus the dominance ratio will be in error. 5) he error in the irst-harmonic eigenvalue o the ission matrix is due to the spatial discretization, may be reduced by reinement o the mesh. However, the corresponding increases in computer memory requirements

2 75 homas M. SUON et al. computational eort severely limit the usability o the FMM or practical applications. It should be noted, however, that the FMM has the desirable characteristic that a rough estimate o the dominance ratio may be computed prior to ission source convergence. 5) his gives a Monte Carlo code with a FMM capability the ability to estimate the dominance ratio during the source convergence process. he Coarse Mesh Projection Method (CMPM) which evolved rom a method based on the use o an autoregressive-moving average (2,) (ARMA(2,)) it was developed to overcome the accuracy limitations o the FMM when used with a small number o large mesh cells. 4) Like the FMM, the CMPM uses a spatial discretization. During the Monte Carlo calculation, quantities are tallied that will later be used to compute an estimate o the noise propagation matrix (NPM). At the completion o the Monte Carlo calculation, the NPM is determined its let (adjoint) eigenvector corresponding to the eigenvalue with the largest modulus is calculated. he time series ormed by taking the inner product o this eigenvector the source vector at each generation is an autoregressive order process with a correlation coeicient equal to the dominance ratio. he CMPM estimates the dominance ratio by computing this correlation coeicient. Unlike the case o the FMM, however, the CMPM generally produces an accurate estimate o the dominance ratio even or very coarse mesh. he only disadvantage o the CMPM (at least in its reported implementation to date) relative to the FMM is that the dominance ratio may not be computed until the ission source has ully converged. 5) his paper introduces a new algorithm termed the Noise Propagation Matrix Method (NPMM) or computing the dominance ratio in Monte Carlo analyses. Like the CMPM, the method uses a determination o the NPM by the Monte Carlo code. he essence o the NPMM is that once an estimate o the NPM is obtained, the dominance ratio is very simply ound as the largest-modulus eigenvalue o the NPM. here are three aspects to the work reported on in this paper. First, it is shown that the mathematical derivation previously used to obtain the expressions governing the practical implementation o the CMPM can be improved upon, thus yielding an improved estimate o the NPM. Second, it is shown that the largest-modulus eigenvalue o this NPM provides an estimate o the dominance ratio that is identical to that which would result rom the time series analysis o the CMPM had the same NPM been used. Finally, it is demonstrated that this method allows a reasonable estimate o the dominance ratio to be determined prior to the convergence o the ission source. his method thus retains the coarse-mesh accuracy computational eiciency o the CMPM while also permitting an on-the-ly determination o the dominance ratio during source convergence. II. he Noise Propagation Matrix As mentioned in Section I, both the CMPM the NPMM use a Monte-Carlo-generated NPM in computing the dominance ratio. he NPM is not tallied directly, but instead is computed rom two source correlation matrices that are tallied during the Monte Carlo calculation. In this section, two dierent expressions are derived that relate the NPM to the correlation matrices. he irst such expression was originally derived by Nease, has been used to implement the CMPM in the MCNP Monte Carlo code. 4,5) he second expression derived or the irst time in this paper has been used in the implementation o the NPMM in the MC2 code. 9) It is shown that the new derivation avoids an approximation used in the original derivation. Implementations o the CMPM the NPMM to-date utilize a rectangular mesh superimposed on the Monte Carlo geometry. A discretized representation o the ission source is obtained by tallying the neutron production rate due to m ission in each o the mesh cells. Let S denote the ission source vector at the conclusion o ission-iteration cycle m, where an element o the vector corresponds to the ission source in one o the cells. It is assumed that m is suiciently large that the source vector can be considered a stationary rom variable. Following Nease Ueki, we decompose the ission source vector as m m N N S S e, () where N is the number o neutrons per cycle, S is the undamental-mode ission source vector normalized to the undamental-mode eigenvalue, e m is a stochastic noise vector that represents the deviation o the cycle-m ission source vector rom its expected value. 3) Denoting the ensemble-averaging process by, Eq. () leads to m S NS S. (2) aking the ensemble average o Eq. () using Eq. (2) yields m e. (3) Nease Ueki show that to lowest order in the noise terms, m e is propagated rom cycle to cycle according to m m m e A e ε, (4) m where A is the NPM ε is a vector representing the noise introduced at cycle m. his latter quantity has the property m ε. (5) In Res. 3) 4), covariance matrices are deined in terms o the noise vector e m. Expressions are then ound relating these covariance matrices to the NPM. o-date, however, practical implementations o the CMPM NPMM in production Monte Carlo codes have used an alternative methodology which utilizes correlation matrices m deined in terms o the ission source vector S. 5) In this paper we only consider this latter methodology. Multiplying Eq. (4) through by N, adding NS to PROGRESS IN NUCLEAR SCIENCE AND ECHNOLOGY

3 On-the-Fly Monte Carlo Dominance Ratio Calculation Using the Noise Propagation Matrix 75 both sides, using the property AS, it can be shown that the equation that governs the propagation o the ission source vector rom cycle to cycle is 4) where m m m S A S η, (6) m m N N η S ε. (7) m Multiplying Eq. (6) through rom the right by S (where denotes transpose) ensemble-averaging results in m m m m m m S S A S S η S. (8) he source correlation matrices are deined as m m L S S (9) m m L S S. () With these deinitions, Eq. (8) can be rearranged to obtain an expression relating the correlation matrices to the NPM A m m L η S L. () Up to this point, the derivation o the expression or the NPM in terms o the source vector is the same as given in Re. 4). In that work, the derivation was continued rom this point by making the approximation m m η S L. (2) Substitution o Eq. (2) into Eq. () then produced Aˆ L L I, (3) where we have introduced the symbol  to represent this approximation to the NPM. his expression or the NPM is the one used or the CMPM implementation in the MCNP code. 4,5) We now show that the approximation given by Eq. (2) can be avoided. Using Eqs. () (7), we have m m m m N N N N η S S ε S e. (4) As shown in Res. 3) 4), the noise introduced in one cycle is uncorrelated with the accumulated noise term rom all previous cycles, i.e. m ( m) ε e. (5) Using Eqs. (2), (3), (5) (5), Eq. (4) may be simpliied to m m η S SS. (6) Note that unlike Eq. (2), this expression does not involve any additional approximation beyond those already employed in arriving at Eq. (4). Substituting Eq. (6) into Eq. () yields the improved expression or the NPM A L SS L. (7) his expression or the NPM is the one used or the NPMM implementation in the MC2 code. Details o that implementation will be given in Section IV. III. Equivalence o the Dominance Ratio as Determined by the CMPM the NPMM In this section, it is shown that given correct consistent ormulations o the NPM, the dominance ratio as determined by the CMPM NPMM algorithms are identical. As will be shown in the Appendix, however, the approximation used to obtain the ormulation o the NPM given by Eq. (3) introduces extraneous terms that alter the eigenvalue spectrum to such an extent that is it not directly usable with the NPMM algorithm. he improved ormulation given by Eq. (7), however, works equally well with either algorithm. It has been shown that the dominance ratio given by the time-series analysis o the CMPM is equivalent to the ratio o two Frobenius inner products, i.e. 4) k k dd : L d Ld, (8) dd : L d Ld where d is the let eigenvector o A corresponding to the largest-modulus eigenvalue a λ, i.e. Ad d. (9) Substituting Eq. (7) into Eq. (9), ater some manipulation we obtain dldss dl. (2) o deal with the second term on the let side o the above equation, we use Eqs. (2) (9) to get ds Ad S N das. (2) N As discussed in Re. 3) AS, hence rom Eq. (2) we obtain ds. (22) Multiplying both sides o Eq. (2) rom the right by d using Eq. (22) we obtain dld dld, (23) hence a o maintain consistency with earlier work, we use a subscript to denote the largest-modulus eigenvalue o the NPM, λ, but use a subscript to denote the largest-modulus eigenvalue o the transport equation, k. VOL. 2, OCOBER 2

4 752 homas M. SUON et al. dld dld. (24) Comparing Eq. (24) with Eq. (8), we see that k, (25) k i.e. the largest-modulus eigenvalue o the Monte Carlo estimate o the NPM is identically the dominance ratio that one obtains rom the CMPM using that same NPM. he importance o this result is that one may compute the dominance ratio simply by using any stard method o inding the largest-modulus eigenvalue o A, that the value so obtained has the same avorable mesh-size properties as that obtained using the CMPM. IV. Implementation o the NPMM in MC2 his section provides a ew details o how the NPMM method or calculating the dominance ratio has been implemented in the MC2 Monte Carlo code. 9) Let m denote the number o cycles run beore the tallying o the elements o the correlation matrix is begun. Ater cycle m ( mm ), the ollowing quantities have been accumulated: m m i i im Lˆ S S, (26) m m i i im 2 Lˆ S S, (27) m m i Sˆ S. (28) im he cycle-m estimate o the NPM is then computed as ˆ m m m ˆm ˆmˆm m A L S S L. (29) mm mm he largest-modulus eigenvalue o m A, hence the cycle-m estimate o the dominance ratio, is ound using the SGEEV subroutine rom LAPACK. ) he deault procedure in MC2 is to make m the same as the number o discard cycles used to converge the ission m source. In this case, A is computed the dominance ratio estimated starting with the second non-discarded cycle. As will be seen in the next section, however, the NPMM seems to provide a reasonable estimate o the dominance ratio well beore the ission source has converged. hus, like the FMM but unlike the CMPM, the NPMM can provide on-the-ly estimation o the dominance ratio during the source convergence process. V. Results. Slab Problem his problem is a one-dimensional (-D), 2-cm-thick able Macroscopic Cross sections or the slab problem slab with vacuum boundary conditions. It assumes isotropic scattering one-group cross sections (provided in able ). he analytic diusion theory solution or the nth-harmonic eigenvalue is kn ; 2 a DBn n,,, (3) where n Bn, L (3) is the geometric buckling, L L2.74 (32) t Cross section Value (cm ). t.7 s.3 a.5.39 able 2 Dominance ratio using ten mesh intervals or the slab problem Reerence NPMM CMPM 4) (6).9986() tr is the slab thickness L plus twice the distance to the extrapolated boundary, tr is the transport mean ree path. For the case o isotropic scattering considered here, tr t D 3. Using Eq. (3), one obtains k k , giving a reerence dominance ratio o An additional reerence solution was generated using the PARISN discrete ordinates code with 5, spatial mesh cells an S 2 quadrature to compute k, then estimating the dominance ratio as k 3 k, where k is the ininite-medium multiplication actor.,2) his procedure also yields a dominance ratio o his problem was analyzed using MC2 with 2, neutrons per cycle, 8, discard cycles 2, active cycles. he NPM was calculated during the 2, active cycles. able 2 gives the reerence dominance ratio, as well as the values obtained using the NPMM the CMPM. 4) he NPMM CMPM calculations were run using ten equally-spaced mesh intervals across the slab. he values in parentheses ollowing the dominance ratio results are two-stard-deviation uncertainties in the least-signiicant digits, where the variance is determined using PROGRESS IN NUCLEAR SCIENCE AND ECHNOLOGY

5 On-the-Fly Monte Carlo Dominance Ratio Calculation Using the Noise Propagation Matrix 753 able 3 Eect o the number o mesh intervals on the dominance ratio or the slab problem Number o mesh intervals NPMM dominance ratio Dierence rom reerence solutions (7) (6) (6) (6).3.999(6) (6).2 4 cm 4 cm t. cm - s.7 cm -.39 cm - t. cm - s.7 cm -.24 cm -. Fig. 2 2-D checkerboard problem geometry nuclear data Dominance Ratio Fig Active Cycle Number Convergence o the dominance ratio or the slab problem 2 2 k 2, (33) M k with M m m being the number o cycles over which the NPM is computed. his ormula was derived or the CMPM based on a result rom time series analysis due to Box Jenkins. 4,3) Since as demonstrated in Section III the NPMM result is identical to the result that one would have obtained rom the CMPM with the same NPM, it ollows that the variance o the NPM result should be the same as that o the CMPM result hence the use o Eq. (33) is appropriate or both methods. he NPMM result agrees well within the statistical uncertainty with both the reerence value the CMPM result. able 3 shows the results or six MC2 calculations where the number o mesh intervals was varied. Even with just two mesh cells, the NPMM result is in agreement with the reerence value. Figure is a plot o the dominance ratio versus active cycle number, along with the 2 b about the inal value. Convergence is initially very rapid, reaching two-signiicant-igure accuracy within about 2 cycles. o attain three-signiicant-igure accuracy, however, requires over,3 cycles. able 4 Dominance ratio results or the 2-D checkerboard problem Method Mesh Dominance ratio Dierence rom discrete ordinates Discrete ordinates.958 FMM CMPM (Nease) (29). CMPM (MCNP) (). NPMM (28) wo-dimensional Checkerboard Problem his problem consists o a two-dimensional (2-D), 6 6 array o two types o square regions arranged in a checkerboard pattern. Vacuum boundary conditions apply to all our sides. he problem employs one-group cross sections isotropic scattering. he geometry cross section data are given in Fig. 2. his problem was analyzed with MC2 using 4 discard cycles 4, active cycles o 8, neutrons each, a 6 6 mesh. able 4 compares the dominance ratio obtained using the NPMM in MC2 with that obtained using the CMPM, a spectral radius analysis o a discrete ordinates calculation, a FMM calculation using a highly-resolved mesh. wo CMPM results are given. he irst is rom a Monte Carlo code used by Nease or his PhD thesis. 4) he NPM used or that calculation is equivalent to that used by the NPMM in MC2 given by Eq. (7). he other CMPM result is rom an MCNP calculation, was obtained using the NPM given by Eq. (3). As can be seen, the NPMM result agrees with the reerence discrete ordinates ine-mesh FMM results at the two-stard-deviation level. Both CMPM results also agree with the reerence calculations at the two-stard-deviation level. Note that although we compare to the discrete ordinates solution, its degree o accuracy is unknown. 3. Large wo-dimensional Problem his problem is a 2-D, 8 8 array o three types o square regions with vacuum boundary conditions. 6) he geometry cross section data are given in Fig. 3. his problem was analyzed using MC2 with, discard VOL. 2, OCOBER 2

6 754 homas M. SUON et al. t s a. cm.7 cm.3 cm.24 cm.3 cm.39 cm able 5 Dominance ratio results or the large 2-D problem Method Mesh Dominance ratio FMM FMM FMM CMPM (MCNP) (2) PARISN (vac. b.c.) PARISN (extrap. b.c.) ARMA(2,) Not applicable.9993(4) NPMM (5).98 Fig. 3 Large 2-D problem geometry nuclear data 4, active cycles o, neutrons each, a 4 4 discretization. he dominance ratio results are given in able 5, along with results obtained using the FMM, the CMPM as implemented in MCNP, two PARISN calculations using a mesh S 6 quadrature, the ARMA(2,) method. 6) Both PARISN calculations estimated the irst harmonic eigenvalue using core symmetry, with one using a vacuum boundary condition the other an extrapolated zero-lux boundary condition on the symmetry plane. he FMM results show the strong dependence o the accuracy o this method on mesh size. It is possible that an even iner mesh than the inest shown here would change the result in the third signiicant digit. hereore, the FMM results do not constitute a reerence calculation, but are merely suggestive. he CMPM result is only given to three signiicant digits, but is in statistical agreement with the NPMM result. Both the ARMA(2,) the NPMM results are given to our signiicant digits. Even though they do not quite agree to within the combined two-stard-deviation uncertainty, they do agree to three signiicant digits. he NPMM result is in good agreement with both PARISN results. Figure 4 shows two plots o the dominance ratio estimate or this problem as a unction o cycle number. In one case, the deault procedure was used in which computation o the NPM dominance ratio is not begun until the discard cycles have been completed the source is ully converged. Note that in this calculation, initial cycles were discarded to minimize the possibility o non-convergence o the source or the purpose o this comparison. In typical calculations, such a large number o discard cycles is not necessary. In the other case, the computation o the NPM dominance ratio is begun immediately with the second discard cycle. he trajectories o the dominance ratio versus cycle number are very similar in the two cases, indicating that the NPMM can yield an accurate estimate o the dominance ratio even i the ission source is not converged. Dominance Ratio Hoogenboom-Martin Problem he inal problem considered is one speciied by Hoogenboom Martin, is an idealized representation o a large power reactor core. 4) his model consists o 24 uel assemblies surrounded by a 2 cm thick steel vessel with an inner radius o 23 cm. Each assembly consists o 7 7 pin positions, is 4 cm high. here is borated water within the assemblies, between the assemblies the reactor vessel, in the regions 2 cm above below the core. his model was used in a calculation that consisted o,, neutrons per cycle,, active cycles, 25 discard cycles. Using a 4 4 mesh, beginning the calculation with the irst active cycle, MC2 computes a dominance ratio or this model o.992(6). o-date no other dominance ratio value or this model has been published, so there is no other result to compare to. We present this simply as an indication that the method gives a reasonable result or a very large, airly realistic, ull-core model using actual nuclear data rather than the one-group, isotropic scattering data used or the other problems. he result also provides a value or uture comparisons. Figure 5 shows the dominance ratio estimate versus cycle or this problem. V. Conclusions dominance ratio calculation begun ater, initial cycles dominance ratio calculation begun with initial lat source guess Cycle Number Fig. 4 Eect o discarded cycles on the dominance ratio calculation A new expression or determining the NPM has been derived that eliminates an approximation that has igured in some o the previous literature on this topic. A new method or computing the dominance ratio in Monte Carlo reactor calculations called the NPMM has been described. In this method, the dominance ratio is computed by inding the PROGRESS IN NUCLEAR SCIENCE AND ECHNOLOGY

7 On-the-Fly Monte Carlo Dominance Ratio Calculation Using the Noise Propagation Matrix 755 Dominance Ratio largest-modulus eigenvalue o the NPM. It is shown that or consistent NPMs, the dominance ratio computed using the NPMM is identical to that computed by the CMPM. he method thus has the desirable characteristic o the CMPM that it is accurate even or a very coarse mesh. It has also been demonstrated that like the FMM, the NPMM has the potential to compute reliable estimates o the dominance ratio during the ission source convergence process. Acknowledgements he authors would like to acknowledge the eorts o Victoria J. Sullivan, Daniel J. Kelly Peter S. Dobre (all o Bechtel Marine Propulsion Corporation Knolls Atomic Power Laboratory) in creating many o the MC2 computational models running the associated calculations reported on in this paper. hey would also like to thank D. Kent Parsons (Los Alamos National Laboratory) or a thorough review o the paper or perorming the PARISN analyses. Reerences Active Cycle Number Fig. 5 Convergence o the dominance ratio or the Hoogenboom-Martin problem ). Ueki, F. B. Brown, Autocorrelation dominance ratio in Monte Carlo criticality calculations, Nucl. Sci. Eng., 45, (23). 2). Ueki, F. B. Brown, D. K. Parsons, ime series analysis o Monte Carlo ission sources I: dominance ratio computation, Nucl. Sci. Eng., 48, (24). 3) B. R. Nease,. Ueki, ime series analysis o Monte Carlo ission sources III: coarse-mesh projection, Nucl. Sci. Eng., 57, 5 64 (27). 4) B. R. Nease, ime series analysis o Monte Carlo neutron transport calculations, Ph.D. dissertation, Dept. o Chemical Nuclear Engineering, Univ. o New Mexico (28). 5) B. Nease, F. Brown,. Ueki, Dominance ratio calculations with MCNP, Proc. Int. opical Mtg. on Physics o Reactors, Interlaken, Switzerl, Sept. 4 9, 28, Paul Scherrer Institut (28), [CD-ROM]. 6). Ueki, B. R. Nease, ime series analysis o Monte Carlo ission sources II: conidence interval estimation, Nucl. Sci. Eng., 53, 84 9 (26). 7) E. M. Gelbard, R. Prael, Computation o Stard Deviations in Eigenvalue Calculations, Prog. Nucl. Energy, 24, (99). 8) K. W. Morton, Criticality Calculations by Monte Carlo Methods, AERE /R 93, Atomic Energy Research Establishment (AERE) Harwell (National Archives o the UK, doc. re. AB 5/589) (956). 9). M. Sutton,. J. Donovan,. H. rumbull, P. S. Dobre, E. Caro, D. P. Griesheimer, L. J. yburski, D. C. Carpenter, H. Joo, he MC2 Monte Carlo code, Proc. Int. Con. on Mathematics & Computation Supercomputing in Nuclear Applications, Monterey, Caliornia, USA, April 5 9, 27, American Nuclear Society (27), [CD-ROM]. ) E. Anderson, Z. Bai, C. Bischo, S. Blackord, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammerling, A. McKenny, D. Sorensen, LAPACK Users Guide (3 rd Ed.), Society or Industrial Applied Mathematics (999). ) R. E. Alcoue, R. S. Baker, J. A. Dahl, S. A. urner, R. Ward, PARISN: a time-dependent parallel neutral particle code system, LA-UR Los Alamos National Laboratory (LANL) (25). 2) D. K. Parsons, D. E. Kornreich, Approximations to the dominance ratio using eective ininite multiplication results, Proc Nuclear Mathematical Computational Sciences: A Century in Review, A Century Anew, Gatlinburg, ennessee, USA, April 6, 23, American Nuclear Society (23), [CD-ROM]. 3) G. E. P. Box, G. M. Jenkins, ime Series Analysis: Forecasting Control, Holden-Day, Oakl, Caliornia (976). 4) J. E. Hoogenboom, W. R. Martin, A proposal or a benchmark to monitor the perormance o detailed Monte Carlo calculation o power densities in a ull size reactor core, Proc. Int. Con. on Mathematics, Computational Methods & Reactor Physics (M&C 29), May 3 7, 29, Saratoga Springs, New York, USA, American Nuclear Society (29), [CD-ROM]. Appendix: he Connection between Various Formulations o the Noise Propagation Matrix In his thesis, Nease derived two alternative ormulations or the NPM. 4) he irst was obtained in terms o the discretized ission source noise vector, is given by where A L L, (A) m m L e e, (A2) m m L e e. (A3) his ormulation involved no approximations beyond those inherent in Eq. (4). His second derivation was developed to acilitate practical implementation in Monte Carlo codes, used the ission source vector itsel rather than the noise vector. his derivation introduced the additional approximation given by Eq. (2). Using the symbol  to distinguish the NPM in this ormulation o the NPM rom that o Eq. (A), we have Aˆ L L I. (A4) A third ormulation has been obtained in this paper, is given by Eq. (7). Like the ormulation given by Eq. (A), it VOL. 2, OCOBER 2

8 756 homas M. SUON et al. involves no additional approximations beyond those used to obtain Eq. (4). Like the ormulation given by Eq. (A4), however, it is given in terms o the ission source vector. For the purposes o this Appendix, this ormulation is denoted A, is given by A L SS L. (A5) Since the ormulations given by Eqs. (A) (A5) were both obtained using Eq. (4) without any additional approximation, they must be equivalent. We will now show this equivalence explicitly. Using Eqs. (), (2), (9), (), (A2) (A3), we may obtain the ollowing relationships between the correlation matrices: L N S S NL (A6) 2 L N S S NL. (A7) 2 Substituting Eqs. (A6) (A7) into Eq. (A5) applying the Sherman-Morrison ormula b yields A A I NSSL NSLS. (A8) Using AS yields A A. he NPM used in the NPMM described in this work is thus identical to the irst one derived in Nease s thesis, but is computed in terms o the source vectors rather than the noise vectors. Since it was derived rom an equation that is accurate to irst order in the noise terms, the eigenvalue spectrum ( hence the dominance ratio) is thus also accurate to that same order. We now turn our attention to the alternative ormulation rom Nease s thesis. Substituting Eqs. (A6) (A7) into Eq. (A4) applying the Sherman-Morrison ormula yields ˆ SSL A A I, (A9) SLS N which in the limit o large N becomes ˆ SSL A A I. (A) SLS he last two terms on the right-h side are due to the additional approximation given by Eq. (2). hese terms which are o the same order in the noise terms as is A itsel cause the eigenvalue spectrum o  to dier signiicantly rom that o A. his ormulation is thus unsuitable or use in the NPMM. b Gene H. Golub Charles F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore (989). PROGRESS IN NUCLEAR SCIENCE AND ECHNOLOGY