Vectorization of Continuous Energy Monte Carlo Method for Neutron Transport Calculation

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1 Journal of NUCLEAR SCIENCE and TECHNOLOGY, 29[4], pp. 325~336 (April 1992). 325 Vectorization of Continuous Energy Monte Carlo Method for Neutron Transport Calculation Takamasa MORI, Masayuki NAKAGAWA and Makoto SASAKIt Japan Atomic Energy Research Institute* Received April 26, 1991 Revised September 13, 1991 The vectorization method was studied to achieve a high efficiency for the precise physics model used in the continuous energy Monte Carlo method. The collision analysis task was reconstructed on the basis of the event based algorithm, and the stack-driven zone-selection method was applied to the vectorization of random walk simulation. These methods were installed into the vectorized continuous energy MVP code for general purpose uses. Performance of the present method was evaluated by comparison with conventional scalar codes VIM and MCNP for two typical problems. The MVP code achieved a vectorization ratio of more than 95% and a computation speed faster by a factor of 8~22 on the FACOM VP-2600 vector supercomputer compared with the conventional scalar codes. KEYWORDS: vectorization, efficiency, computer codes, continuous energy, Monte Carlo method, neutron transport, computer calculations I. INTRODUCTION The continuous energy Monte Carlo method is widely used as a powerful tool to obtain a reference solution or to analyze experiments as precisely as possible in the field of particle transport problems. The Monte Carlo method, however, usually requires much computation time to obtain a result with required statistics due to its statistical nature. Furthermore, the continuous energy method consumes more computation time by about 30~80% owing to the accurate treatment of physics models compared with the multi-group one. The speedup of the continuous energy codes would extend the applicability and benefit. An advent of recent supercomputers with vector processors reduces computation costs by vectorization of conventional scalar codes. For Monte Carlo codes, however, the vectorization is not a simple work but it needs an almost complete rewrite of a program. Conventional scalar codes use the history based algorithm, and they are impossible to be vectorized. It was demonstrated that Monte Carlo calculations could be successfully vectorized by adopting the event based algorithm(1)~(5). However, there are large differences in individual approaches for vectorization and they significantly affect a performance of vectorized code(6). In addition, a vectorization efficiency and an optimum algorithm depend on the physical nature of the problem to be solved and the computer architecture. It is important to investigate an appropriate algorithm and a portability of vectorized codes between different supercomputers for the development of general purpose Monte Carlo codes. In Japan Atomic Energy Research Institute, such a study was extensively carried out and a vectorized multi-group Monte Carlo code GMVP for general purpose uses was developed on the FACOM VP-100(7)~(9). Furthermore, t Tokai-mura, Ibaraki-ken * Permanent address : Japan Research Institute Ltd., Kami-Ohsaki, Shinagawa-ku, Tokyo 141,, 23

2 326 J. Nucl. Sci. Technol., the experience of installing the GMVP code on other supercomputers indicated that the vectorized Monte Carlo code was portable and a satisfactory performance could be achieved on fairly different supercomputers by the conversion with modest efforts(7)(9). It was also found that the speedup depended on the number of vector pipelines installed in these computers. The detailed treatment of individual interactions and the handling of a large amount of nuclear data introduce an increasing complexity into the vectorization of the continuous energy method compared with the multi-group one. Brown et al. adopted a rather simple physics model of neutron interactions based on the 05R code in their vectorized code LACER3D by restricting its application to reactor calculations(5). In the present work, we studied the vectorization method for general purpose codes which require accurate physics models for wide applications as much as possible. Several physics models suitable for vector computations were established and are proposed in Chap. II together with sampling methods from them. Furthermore, the control logic of complicated analysis of interactions and the cross section data management were studied to achieve a high vectorization efficiency for precise Monte Carlo calculations. Those are described in Chap. III. By combining these procedures with the stack-driven zone-selection method developed for the GMVP code(8), a vectorized continuous energy code MVP was developed for general purpose uses with the FACOM VP-2600 vector supercomputer. To evaluate a performance of the present method, the MVP code was compared with conventional scalar codes VIM(10) and MCNP(11) for two typical problems. The results are presented in Chap. IV, where the vectorization efficiency is estimated by absolute computation speed (CPU time/track) and speedups. Furthermore, an amount of vectorized part of the code is measured by the vectorization ratio on the CPU time basis which sets an upper limit of speedup due to vectorization. II. PHYSICS MODEL AND SAMPLING TECHNIQUES FOR VECTORIZED CONTI- NUOUS ENERGY METHOD 1. Expression of Cross Sections and Probability Data In the present work, we treat the energy range of 20 MeV~10-5 ev. All neutron reaction channels given in the evaluated nuclear data file for this energy range are explicitly taken into account. To ensure a realistic representation of the physics model and an efficient and accurate numerical procedure, the evaluated nuclear data of each nuclide are processed into an appropriate library. Neutron cross sections in the resolved resonance region and smooth one are pointwisely represented, and those at any intermediate energy are obtained by a linear-linear interpolation formula. Energy points at which cross sections are given are selected to reproduce cross sections within a required accuracy (normally 0.1%) by the interpolation. In the unresolved resonance region, cross sections are obtained by the cross section probability table method. The probability table consists of several sets of values for si, se and sf and their probabilities p. With a probability p, these cross sections take specified values. For the sampling from the probability table, the code uses the discrete conditional sampling method(14), which needs no IF-test and is very fast regardless of the number of possible events. Probabilities such as energy and angular distributions of secondary neutrons are given by either tables or specific functions with energy-dependent parameters. Both tables and parameters are given at a number of different incident energies, depending on nuclides and reactions. In the former case, a table to be sampled from is probabilitically selected. When En and En+1 are the energies of the n-th and (n+1)-th tables, respectively, the (n+1)-th table is selected with a probability p= (En-E)/(En-En+1), and the n-th table with a probability 1-p for En+1<=E<=En. This selection is made without an IF-test by find- 24

3 Vol. 29, No. 4 (Apr. 1992) 327 ing the largest integer which is less than or equal to n+p+x, where is a random number uniformly distributed in the interval (0, 1). On the other hand, a value of parameter at any energy is calculated by the linear-linear interpolation. 2. Analysis of Neutron Interaction The types of neutron interactions are summarized in Fig. 1, and the collision analysis proceeds as shown in Fig. 2. The physics Fig. 1 Neutron interactions in MVP code Fig. 2 General flow of collision analysis 25

4 328 Nucl. Sci. Technol., models of neutron interactions and the sampling methods adopted to increase a vectorization efficiency are briefly described below. (1) Sampling of Collision Nuclide A neutron collides with a nuclide h in a zone composed of K nuclides with a probability Pk: where r, st, and St are the atomic number density, the microscopic and macroscopic total cross sections, respectively. Conventional scalar codes use cumulative probabilities to sample a collision nuclide. On the other hand, the present code determines a collision nuclide by a new method which uses K random numbers xk in the interval (0, 1). If rkskt/ =1,rjsjt>xk, the k-th nuclide is reserved Skj as a candidate. This test is continued from the first constituent nuclide to the last one. The last nuclide reserved is selected as a collision nuclide. (2) Creation of Fission Neutrons A constant weight Wf is assigned to each fission neutron, and the number of created neutrons n is probabilitically determined to preserve the expected value <n>: where Wnsf/st is the expected value of total weight of fission neutrons created at each collision. The method to determine n is the same as that to select a table to be sampled from, which is described in Chap. II-1. The quantity n+p described there is replaced with <n> in this case. (3) Treatment of Absorption Two methods are used to treat absorption of neutrons, as shown in Fig. 1. One is the analog absorption used in a lower energy region where a neutron history is terminated with an absorption probability sa/st. The other is the implicit absorption treatment (nonanalog absorption) used in a higher energy region. In this energy region, a neutron weight is reduced by a factor of a nonabsorption probability instead of terminating a history. A boundary of two energy regions is set to be 275 ev as a default value, but it can be optionally changed to increase a calculation efficiency in particular problems. A lower energy is generally preferable for problems with highly absorbing materials from the viewpoint of increasing statistics in the low energy region. On the other hand, a higher energy is used to suppress the excessively large number of collisions and to reduce computation time for problems with a low absorption probability. (4) Determination of Reaction Type of Scattering The type of scattering process is determined by sampling from the probabilities px=sx/(st-sa), where x indicates a reaction type. This sampling is performed by a linear search on a cumulative probability table to find l such as If the (n, mn) reaction is selected, (m-1) neutrons are created. (5) Elastic and Inelastic Scattering (a) Sampling of scattering angle The scattering angle m is sampled in the same manner for all elastic and inelastic scattering processes. The angular distribution tables consist of equiprobable cosine bins (mi) and the cosines are either in the center-ofmass or in the laboratory system, depending on the type of reaction. The sampling process proceeds by using two random numbers and x2 as follows : The i-th cosine x1 bin is selected without an IF-test by uniform sampling such as i-1<=x1n<i, where N is the number of equiprobable bins. Then, the value of m is calculated as m=mi+x2(mi+1-mi) by assuming an isotropic distribution within each bin. (b) Sampling of secondary energy For the elastic and inelastic scattering from discrete levels, the secondary energies are determined by the two-body kinematics. On the other hand, those from other inelastic scattering processes are governed by various scattering laws. The MVP code can treat all 26

5 Vol. 29, No. 4 (Apr. 1992) 329 laws used in the JENDL-3 evaluation : (1) arbitrary tabulated function, (2) general evaporation spectrum, (3) simple fission spectrum, (4) evaporation spectrum and (5) energy dependent Watt spectrum. For the first two laws, equiprobable bins are adopted. The sampling processes from these laws are the same as that of scattering angles except that the probability distribution in each bin is not constant but a linear function specified by given probabilities. Other three laws are well-known functions. To sample from these functions, the code uses the direct methods without an IF-test presented in Ref. (12). (6) Thermal Scattering In the thermal energy region, neutron scattering is treated based on the free gas, the infinite mass target, the elastic or the inelastic scattering (S(a,b) law) model, as shown in Fig. 1. The third model corresponds to the elastic scattering in the ENDF/B thermal library. The S(a,b) data are converted into the energy transfer probabilities and the angular distributions. To find the exiting energy bin, the discrete conditional sampling method is used. Angular distributions are specified for each energy transfer by the MARC method(13). 3. Eigenvalues The eigenvalue (keff) is calculated by two methods. One is based on the multiplication of source neutrons defined as The other is based on the neutron balance : where Sloss=Sa-Sn,2n-2Sn,3n-3Sn,4n. terms including cross sections are evaluated by three different estimators : the track length, The the collision and the analog estimators. The first two estimators use macroscopic cross sections, while the last one uses microscopic cross sections of selected collision nuclides. As a results, we obtain six different estimates of keff, which correlate with each other. Then, the MVP code calculates the best estimate and its variance based on the method of maximum likelihood. Those are obtained as where Xj and Tij are each estimate of keff and an element of the inverse matrix of covariance evaluated from deviation of batchwise estimates by six estimators, respectively. III. VECTORIZATION METHOD In order to vectorize a Monte Carlo simulation, the MVP code adopts the stack-driven zone-selection method which is a variation of the standard event-based stack-driven algorithm. In this method, the simulation process consists of a source particle generation task and five other basic tasks : free flight analysis, next zone search, collision analysis, lattice and reflection treatments. This method was developed for the multi-group code GMVP(8). For spatial tracking of particles, the MVP code utilizes most routines developed for the multi-group code. However, the collision analysis and cross section handling routines are quite different. The different types of neutron interactions are separately processed as described in Chap. II. Furthermore, calculations of cross sections and retrieval of appropriate probability distributions are necessary during a random walk simulation. These require a new control logic of the collision analysis and fast algorithms for table searches and samplings from different probabilities which are suitable for vector supercomputers. Those are described in this chapter. 1. Cross Section Data Management All cross sections and probability data are stored in a single array and the retrieval of each data is performed by using pointers. 27

6 330 J. Nucl. Sci. Technol., Both microscopic and macroscopic cross sections are calculated at any neutron energy from these data during the random walk simulation. The calculated cross sections are referred in different calculation tasks. Furthermore, the use of the cross section probability method for the unresolved resonances requires simultaneous calculations of st, se and sf which are given probabilitically. The code constructs an array called "sigma bank" to store calculated cross sections. The structure of the sigma bank is similar to that of the particle bank. At the end of the source generation and. The mixing operation is simple since II all neutrons processed in this task belong to a single zone consisting of the same material. The code stores macroscopic cross section data of several materials for each neutron in the sigma bank. This capability is useful for problems composed of low density materials where neutrons move through several materials without a collision. 2. Collision Analysis The collision analysis consists of the six steps as shown in Fig. 2. The collision task performs the second to the last steps because the collision nuclide is already selected in the free flight task and stored in the sigma bank. The second and third steps (creation of fission neutrons and treatment of absorption) are carried out in the same way as in the multi-group method by using microscopic cross sections. The scattering analysis is much more complicated compared with the multi-group method. The number of possible reaction channels the collision analysis tasks, microscopic cross sections st, se, sc, sf and nsf are calculated is more than fifty and dependent upon collision nuclides and neutron energies. To pro- for all nuclides, and stored in the sigma bank together with pointers indicating the energy cess these reactions, the code constructs nine grids. These pointers are referred to calculate cross sections for other reactions when This figure shows the flow of neutrons among temporary working stacks shown in Fig. 3. they are required in the course of the execution of collision task. For eigenvalue calcu- divided into three stacks depending on their the stacks. At the first stage, neutrons are lations, a cross section sioss=sc+sf-sn,2n-2sn,3n-3sn,4n energies : thermal scattering #1, #2 and fast is also calculated and stored scattering (see Fig. 1). Each stack stores particle pointers and collision nuclides. The third in the sigma bank. Macroscopic cross sections are calculated stack also stores reaction channels selected. in the free flight task. The collision nuclide The operations for these three stacks are as is determined through the course of mixing follows : operations by the method described in Chap. Thermal #1: Pointers of neutrons are scattered into two stacks by selecting the type of scattering process. Then, a table search is performed to find the table to be used for sampling the secondary energy (inelastic by S(a,b) law) and the scattering angle (thermal elastic). Thermal #2: Neutrons are sorted into two Fig. 3 Connection of temporary working stacks used in collision analysis 28

7 Vol. 29, No. 4 (Apr, 1992) 331 stacks according to the mass of collision nuclide. Fast: The type of scattering reaction is sampled at first. Then, sampling from angular distributions is carried out. After that, pointers of neutrons are moved to two stacks depending on the type of scattering process. The remaining six stacks are processed by vector operations on the basis of the physics models described in Chap. II. As a result, the secondary energy and the flight direction are determined for neutrons in the stack of free gas model, and the secondary energy and the scattering angle cosine for other neutrons. The latter neutrons are gathered and the flight direction is calculated. After that, all neutrons are gathered and the calculation proceeds to the next task. The gathering and scattering operations used here can increase the vector length and the computation efficiency due to vectorization. 3. Vectorization of Table Search Table searches are necessary for several purposes. Two methods, binary and linear searches, are generally used in a Monte Carlo code. In the present code, the binary search is used to find an incident energy grid for retreaval of appropriate data and to find an energy group corresponding to each neutron energy for tallies. The binary search is especially efficient for the case of large size tables. On the other hand, the linear search is used for sampling from a cumulative probability table. Cumulative probabilities are calculated through the course of random walk simulation as in the sampling of types of scattering processes. Probability data are given at different incident energy grids depending on nuclides and reaction types. The energy grids to be searched for each neutron are stored in different tables. Accordingly, a capability of simultaneous searches of various tables with different length is essential to obtain a high computation gain. To implement this capability, we have developed a new scheme for binary searches which uses pointers indicating different tables together with Brown's method for a single table(2). Simultaneous linear searches for many neutrons are vectorized by the same approach as that for loops containing a feed-backward type IF-test(15) which appears in the rejection sampling method. The vectorized scheme is shown together with the scalar one in Fig. 4. The backward path depicted in the left side Fig. 4(a),(b) Vectorization of loops containing feed-backward type IF test (linear search from tables) 29

8 332 J. Nucl. Sci. Technol., of each scheme corresponds to the loop over elements of tables (table length) in the linear search. Two temporary buffers, the Accept and Reject buffers, are defined in the vectorized scheme. The neutrons which satisfy the IF-test are compressed into the former buffer, while the rejected ones are into the latter at the end of each loop. The Reject buffer is processed in the subsequent loop. This process is repeated until the Reject buffer becomes empty. In the vectorized scheme, the feedback path still exists, but the operations 1 and 2 are vectorized with the vector length of the number of neutrons. 4. Particle Tracking Algorithm The random walk simulation is resolved into six tasks, as mentioned above. The connection of these tasks is the same as those in the multi-group code and described elsewhere(8). Each of the five basic tasks has its own stack for queueing up pointers of particles and for storing attributes of the stack required in a selection of the next task to be executed. The order of processing tasks is not fixed but depends on the numbers of queued particles in the stacks (stack-driven algorithm). All particle descriptors reside in a large particle bank, and some of them are gathered and arranged into working arrays for vector operations in each task. After the vector operation, the updated particle descriptors are scattered back to the bank. In the zone-selection method which is a standard algorithm of the present code, two tasks of the free flight analysis and the next zone search are carried out for particles in a single geometric zone. In three other tasks, all particles in the stack are processed regardless of zone to which each particle belongs. In most cases, the chance of processing these three tasks is fairly small compared with the first two tasks in order to maximize the vectorization efficiency. The method to select the next task is as follows : When the number of particles queued for the collision, lattice or reflection task exceeds a preset value (i.e. half of the number of active particles), that task is selected. For other two tasks, the zone with the most particles is found out for each task at first and then the task with the most particles is selected among those. The detail is described elsewhere(9). 5. Main Capability of MVP Most of functions for production use are implemented in the code. The current capabilities of the MVP code are summarized as follows : (1) Problem to be solved : fixed source and eigenvalue keff problems of neutron transport. (2) Description of geometry : combinatorial geometry with multiple square and hexagonal lattices. Available unit bodies : hemispace, right parallelepiped, arbitrary polyhedron, right circular cylinder, sphere, truncated right cone, ellipsoid, triangular prism, hexagonal prism and torus. Boundary conditions : vacuum, perfect and white reflections. (3) Tallies : flux, reaction rate, averaged macroscopic and/or microscopic cross sections, keff and variances of these estimates. (4) Variance reduction techniques : Russian roulette kill and splitting based on cell importance or weight window. IV. PERFORMANCE EVALUATION OF MVP CODE To evaluate the efficiency of vectorization, two typical problems were solved by the MVP code. One is an eigenvalue problem of a PWR fuel assembly(8) and the other is a shielding problem of deep penetration of 14 MeV neutrons(16). 1. PWR Fuel Assembly As a fission source problem, a three-dimensional PWR fuel assembly was calculated with 17 x 17 pins including fuel, cladding, 24 control rod channels and a channel for instrumentation(8). The model is shown in Fig. 5. This is a typical lattice geometry problem. The reflector regions composed of iron and water are placed at the top and bottom ends of the assembly. The total number of nuclides is six in this problem. To estimate a computation gain due to vectorization, the result and performance are 30

9 Vol. 29, No. 4 (Apr. 1992) 333 scalar mode calculation)-(cpu time for scalar operations in a vector mode calculation)}/(total CPU time in a scalar mode calculation)) is about 95%. When the batch size is increased to 20,000, a speedup of a factor of 9.7 is obtained. If the tally calculation for cross section edit is suppressed, the speedup and the vectorization ratio increase by 60 and 3%, respectively, as shown in the parentheses of Table 1. This is due to the fact that most of the tally calculation cannot be vectorized on current vector supercomputers and must be carried out by scalar operations. Table 1 Performance comparison for PWR fuel assembly problemt1 Fig. 5 Side view of PWR fuel assembly compared with those of the conventional scalar code VIM in Table 1. The total number of particles is the same for both codes and is 100,000. The eigenvalues keff calculated by these two codes agree with each other within their fractional standard deviations (FSD), but the FSD value of MVP evaluated by six different estimators is smaller than that of VIM which uses three estimators. As a measure of computation speed, we use a CPU time/ track, where the number of tracks is the summation of the collision and boundary crossing events. This quantity does not strongly depend on the problem to be solved but depends on the used vectorization method, computer and vector length(6). The speedup (CPU/ track ratic of the VIM to MVP calculations) of a factor of 8.1 is achieved with a batch size of 5,000 particle/batch on the FACOM VP The vectorization ratio ({(total CPU time in a The performance and vectorization efficiency of each basic task are shown in Table 2. The vectorization efficiency is presented in terms of a speedup due to vector operations (a ratio of CPU time used in a scalar mode calculation to that in a vector one). In this problem, the computation time is mainly consumed by the free flight and collision tasks. The frequency of collision analysis is smaller by one order compared with the free flight task. The speedup of the collision analysis task is a factor of 13.6 while the highest and overall ones are factors of 33.0 (in the next zone search task) and 12.2, respectively. It should be noted that the free flight task consumes more than 60% of the computation time in tallying reaction rates for the cross section edit by the track length estimator. Without the cross section edit, the relative CPU time decreases to 36%, and the speedup of this task becomes a factor of

10 334 J. Nucl. Sci. Technol., Table 2 Performance and vectorization efficiency of MVP in PWR fuel assembly problemt1 Table 3 compares the performance of the collision analysis between the continuous energy method (MVP) and the multi-group one (GMVP). In this table, the results without the cross section edit are presented. In this case, the two codes show a similar performance in the tasks except for the collision analysis. The MVP code consumes a 6 times longer CPU time in the collision analysis than GMVP, and relative CPU times for this task are 50 and 15%, respectively. The increment of total CPU time due to the continuous energy model is a factor of 1.8. However, comparable speedups are achieved by both codes. When the number of nuclides in the problem is increased, more CPU time is required for the calculation of microscopic and macroscopic cross sections which includes the selection of collision nuclides. For such a problem, however, a comparable speedup can be expected since the vectorization efficiencies for these calculations are unchanged from those in Table 3. From the present results, it can be said that a satisfactory vectorization efficiency is realized for the collision analysis by the present method. Table 3 Comparison of collision analysis in PWR fuel assembly problem between continuous energy and multi-group methods (MVP and GMVP)t1 2. Shielding Problem of Deep Penetration As an example of a fixed source problem, we solved a deep penetration problem through an infinite slab of iron or iron and concrete with 3 m thickness. The calculation model is shown in Fig. 6. The problem for iron was proposed by Hendricks et al. as a computational benchmark(16). In the latter case, 10 cm thick iron and concrete slabs are repeatedly placed. The infinite slab was modelled by using perfect reflecting surfaces in threedimensional geometry. A mono-directional 14 MeV neutron source is located on the one side of the infinite slab, and the importance assigned to each geometrical zone is widely varied from unity at the source to ~109 at 3 m depth. The importance is increased by a factor of 2 every 10 cm in order to enhance the deep penetration. Twelve detector regions of 1 cm thickness are used to evaluate spatial 32

11 Vol. 29, No. 4 (Apr. 1992) 335 Fig. 6 Calculation model of deep penetration problem Table 4(a),(b) Performance and vectorization efficiency of MVP in deep penetration problemt1 variation of the neutron flux spectrum by the track length and collision estimators. The detailed performance is summarized in Table 4(a),(b). The speedups in the collision task (14.8 and 13.4) are similar to that obtained in the eigenvalue problem, although these values are slightly lower than the overall speedups (17.2 and 15.3). Accordingly, the present vectorization method used in the collision analysis is efficient in wide application. The absolute performance and the calculated values are compared with the MCNP code in Table 5. The results by these codes show a good agreement for neutron leakage and slowing-down below the cut-off energy. The computation speed of MVP is higher by factors of 22 and 16 for two cases than MCNP. The vectorization ratio is about 98 % in this Table 5 Performance comparison for 3 m thick deep penetration problemt 33

12 336 J. Nucl. Sci. Technol., problem since no tally calculation is made for cross section edits. V. CONCLUSION The vectorization method of the continuous energy Monte Carlo method has been studied for neutron transport calculations. Based on the algorithm developed, the vectorized continuous energy MVP code has been developed for general purpose uses on the FACOM VP-2600 vector supercomputer. The present code achieved a vectorization ratio of more than 95 %, which means that more than 95 % of computation time consumed in a scalar calculation is processed by vector operations. The speedups of factors of 8~22 were achieved compared with the conventional scalar codes VIM and MCNP. The present work has demonstrated that a computation gain by vectorization of a general purpose code can significantly reduce a computation cost and would enhance the use of continuous energy Monte Carlo method in wide applications. For further extension of application, the studies are under way for capabilities of the photon transport calculations and point detector estimation. REFERENCES (1) BROWN, F. B. : Trans. Am. Nucl. Soc., 43, 377 (1982). (2) idem: Vectorized Monte Carlo methods for reactor lattice analysis, Proc. Am. Nucl. Soc. Top. Mtg. on Advances in Reactor Computations, Salt Lake City, Ma, 1983, p (3) BROWN, F. B., et al.: Prog. Nucl. Energy, 14, 269 (1984). (4) BOBROWICZ, F. W.: Parallel Comput., 1, 298 (1984). (5) BROWN, F. B.: Trans. Am. Nucl. Soc., 53, 283 (1986). (6) MARTINE, W. R., BROWN, F. B.: IBM J. Res. Develop., 30, 193 (1986). (7) NAKAGAWA, M., MORI, T., SASAKI, M.: Development of Monte Carlo code for particle transport calculation on vector processor, Proc. Supercomputing in Nuclear Application, Mito, Mar. 1990, p (8) NAKAGAWA, M., MORI, T., SASAKI, M.: Nucl. Sci. Eng., 107, 58 (1991). (9) idem: Prog. Nucl. Energy, 24, 183 (1990). BLOMQUIST, R.N., LELL, R.M., GELBARD, (10) E.M. : A review of the theory and application of Monte Carlo method, ORNL/RSIC-44, p. 31 (1980). (11) BRIESMEISTER, J. B. (ed.) : MCNP-A general Monte Carlo code for neutron and photon transport version 3A, LA-7396-M. Rev. 2, (1986). (12) EVERETTE, C. J., CASHWELL, E. D.: A Monte Carlo sampler, LA-5061-MS, (1972). (13) SPANIER, J., GELBARD, E. M.: "Monte Carlo Principles and Neutron Transport Problem", (1969), Addison-Wesley Publ., Reading Mass. (14) BROWN, F.B., MARTINE, W.R., CALAHAN, D.A.: Trans. Am. Nucl. Soc., 39, 755 (1981). MIURA, K. : Vectorization and (15) parallelization of transport Monte Carlo simulation, Proc Winter Simulation Conf., New Orleans, Dec. 1990, p HENDRICKS, (16) J.S., CARTER, L.L. : Nucl. Sci. Eng., 77, 71 (1981). 34