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  • Abstract - A Parallel Algorithm for Random Walk Construction with Application to the Monte Carlo Solution of Partial Differential Equations
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A Parallel Algorithm for Random Walk Construction with Application to the Monte Carlo Solution of Partial Differential Equations
March 1993 (vol. 4 no. 3)
pp. 355-360

Random walks are widely applicable in statistical and scientific computations. Inparticular, they are used in the Monte Carlo method to solve elliptic and parabolic partialdifferential equations (PDEs). This method holds several advantages over other methodsfor PDEs as it solves problems with irregular boundaries and/or discontinuities, givessolutions at individual points, and exhibits great parallelism. However, the generation ofeach random walk in the Monte Carlo method has been done sequentially because eachpoint in the walk is derived from the preceding point by moving one grid step along arandomly selected direction. A parallel algorithm for random walk generation in regular as well as irregular regions is presented. The algorithm is based on parallel prefixcomputations. The communication structure of the algorithm is shown to ideally fit on ahypercube of n nodes, where n is the number of processors.

[1] V. C. Bhavsar, "Some parallel algorithms for Monte Carlo solutions of partial differential equations," inAdvances in Computer Methods for Partial Differential Equations, vol. 4, R. Vichnevestky and R. S. Stepleman, Eds. New Brunswick: IMACS, 1981, pp. 135-141.
[2] V. C. Bhavsar and V. V. Kantkar, "A multiple microprocessor system (MMPS) for the Monte Carlo solution of partial differential equations," inAdvances in Computer Methods for Partial Differential Equations, vol. 2, R. Vichnevestky, Ed. New Brunswick: IMACS, 1977, pp. 205-213.
[3] V. C. Bhavsar and A. J. Padgaonkar, "Effectiveness of some parallel computer architectures for the Monte Carlo solution of partial differential equations," inAdvances in Computer Methods for Partial Differential Equations, vol. 3, R. Vichnevestky and R. S. Stepleman, Eds. New Brunswick: IMACS, 1979, pp. 259-264.
[4] G. Bilardi and F. P. Preparata, "Size-time complexity of Boolean networks for prefix computations,"J. ACM, vol. 36, no. 2, pp. 362-382, Apr. 1989.
[5] G. E. Blelloch, "Scans as primitive operations,"IEEE Trans. Comput., vol. 38, no. 11, pp. 1526-1538, Nov. 1989.
[6] J. H. Curtiss, "Sampling methods applied to differential and difference equations," inProc. Seminar Scientif. Computat., IBM 1949.
[7] O. Egecioglu, E. Gallopoulos, andÇ. Koç, "Fast and practical parallel polynomial interpolation," Tech. Rep. 646, Center for Supercomputing Research and Development, Univ. Illinois at Urbana-Champaign, Jan. 1987.
[8] J. H. Halton, "A retrospective and prospective survey of the Monte Carlo method,"SIAM Rev., vol. 12, Jan. 1970.
[9] J. M. Hammersly and D. C. Handscomb,Monte Carlo Methods. London, England: Methuen, 1964.
[10] P. M. Kogge and H. S. Stone, "A parallel algorithm for the efficient solution of a general class of recurrence equations,"IEEE Trans. Comput., vol. C-22, no. 8, pp. 786-793, Aug. 1973.
[11] C. P. Kruskal, L. Rudolph, and M. Snir, "The power of parallel prefix,"IEEE Trans. Comput., vol. C-34, pp. 965-968, 1985.
[12] R. E. Ladner and M. J. Fischer, "Parallel prefix computation,"J. ACM, vol. 27, no. 4, pp. 831-838, Oct. 1980.
[13] S. Lakshmivarahan, C.-M. Yang, and S. K. Dhall, "On a class of optimal parallel prefix circuits with (size + depth) = 2n- 2 and⌈logn⌉≤depth≤(2⌈logn⌉- 3)," inProc. Int. Conf. Parallel Processing, Aug. 1987, pp. 58-63.
[14] E. Sadeh and M. A. Franklin, "Monte Carlo solutions of partial differential equations by special purpose digital computers,"IEEE Trans. Comput., vol. C-23, pp. 389-397, Apr. 1974.
[15] M. Snir, "Depth-size tradeoffs for parallel prefix computation,"J. Algorithms, vol. 7, pp. 165-201, June 1986.

Index Terms:
Index Termsparallel algorithm; random walk construction; Monte Carlo solution; partial differential equations; irregular boundaries; discontinuities; randomly selected direction; parallel prefix computations; communication structure; hypercube; hypercube networks; Monte Carlo methods; parallel algorithms; partial differential equations
Citation:
A. Youssef, "A Parallel Algorithm for Random Walk Construction with Application to the Monte Carlo Solution of Partial Differential Equations," IEEE Transactions on Parallel and Distributed Systems, vol. 4, no. 3, pp. 355-360, March 1993, doi:10.1109/71.210818
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