Abstract
Three Monte Carlo methods for Matrix Inversion (MI) are considered: with absorption, without absorption with uniform transition frequency function, and without absorption with almost optimal transition frequency function.
Recently Alexandrov, Megson and Dimov has shown that an n×n matrix can be inverted in 3n/2 + N + T steps on regular arrays with O(n 2 NT) cells. A number of bounds on N and T have been established (N is the number of chains and T is the length of the chain in the stochastic process, which are independent of matrix size n), which show that these designs are faster than the existing designs for large values of n.
In this paper we take another implementation approach, we consider parallel Monte Carlo algorithms for MI on MIMD environment, i.e. running on a cluster of workstations under PVM. The Monte Carlo method with almost optimal frequency function performs best of the three methods as it needs about six-ten times less chains for the same precision.
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References
Bertsekas D.P. and Tsitsiklis, Parallel and Distributed Computation, Prentice Hall, 1989
Curtiss J.H., Monte Carlo methods for the iteration of linear operators. J. Math Phys., vol 32, No 4 (1954), pp. 209–232.
Curtiss J.H. A Theoretical Comparison of the Efficiencies of two classical methods and a Monte Carlo method for Computing one component of the solution of a set of Linear Algebraic Equations., Proc. Symposium on Monte Carlo Methods, John Wiley and Sons, 1956, pp.191–233.
Dimov I. and O. Tonev, Random Walk on Distant Mesh Points Monte Carlo Method. Journal of Statistical Physics, Vol. 70, Nos.5/6, pp. 1333–1342, 1993.
Delosme J.M., A parallel algorithm for the algebraic path problem Parallel and Distributed Algorithms, eds. Cosnard et all, Bonas, France, 1988, pp67–78.
Ermakov S.M. Method Monte Carlo and related topics, Moscow, Nauka, 1975.
El-Amawy A., A Systolic Architecture for Fast Dense Matrix Inversion, IEEE Transactions on Computers, Vol. C-38, No3, pp. 449–455, 1989.
Holton J.H., A Retrospective and Prospective Survey of the Monte Carlo Method, SIAM Rev., Vol. 12, No1, 1970, pp. 1–63.
Kung S.Y., S.C. Lo and P.S. Lewis, Optimal Systolic Design for the Transitive Closure and shortest path problems, IEEE Transactions on Computers, Vol. C-36, N05, 1987, pp. 603–614.
Megson G.M., A Fast Faddeev Array, IEEE Transactions on Computers, Vol. 41, N012, December 1992, pp. 1594–1600.
G.M.Megson, V.Aleksandrov, I.T. Dimov A Fixed Sized Regular Array for Matrix Inversion by Monte Carlo Method, In Advances in Numerical Methods and Applications, World Scientific, pp.255–264, 1994.
S. Lakka Parallel Matrix Inversion Using a Monte Carlo Method MSc. Dissertation, University of Liverpool, September 1995.
G.M. Megson, V. Aleksandrov, I. Dimov Systolic Matrix Inversion Using Monte Carlo Method, J. Parallel Algorithms and Applications, Vol. 3, pp. 311–330, 1994.
Robert Y. and D. Trystram, Systolic Solution of the Algebraic Path Problem, in Systolic Arrays, W.Moore et. al, Adam Hilger, 1987 pp.171–180
Rajopadhye S., An Improved Systolic Algorithm for the Algebraic Path Problem, Integration the VLSI journal, 14, pp279–290, 1993.
Sobol' I.M. Monte Carlo numerical methods. Moscow, Nauka, 1973 (Russian) (English version Univ. of Chicago Press 1984).
Snell J.L. Introduction to Probability, Random House, New York, 1988
Westlake J.R., A Handbook of Numerical Matrix Inversion and Solution of Linear Equations, John Wiley and Sons, New York, 1968.
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© 1996 Springer-Verlag Berlin Heidelberg
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Alexandrov, V.N., Lakka, S. (1996). Comparison of three Monte Carlo methods for Matrix Inversion. In: Bougé, L., Fraigniaud, P., Mignotte, A., Robert, Y. (eds) Euro-Par'96 Parallel Processing. Euro-Par 1996. Lecture Notes in Computer Science, vol 1124. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0024687
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