General Sequential Sampling Techniques for Monte Carlo Simulations: Part I—Matrix Problems
We study sequential sampling methods based principally on ideas of Halton. Such methods are designed to build information drawn from early batches of random walk histories into the random walk process used to generate later histories in order to accelerate convergence. In previously published work, such methods have been applied within the pseudorandom, rather than the quasirandom, context and have been applied only to matrix problems. In this paper, more general sequential techniques are formulated in an abstract space, such as Banach space. The more general formulation enables applications to linear algebraic equations and to integral equations to be obtained as special cases through specification of the Banach space and the operator defined on it. In this initial paper we outline the ideas needed for consideration of the more general problem and exhibit greatly accelerated convergence for a simple matrix problem. In a companion paper in which similar ideas are applied to the more important class of integral equations, the need for quasirandom implementation is stressed.
KeywordsRandom Walk Importance Sampling Sequential Sampling Transport Problem Actual Error
Unable to display preview. Download preview PDF.
- J.H. Halton, “Sequential Monte Carlo”, Proc. Camb. Phil. Soc. 58, (1962).Google Scholar
- J. Halton, “Sequential Monte Carlo Techniques for the Solution of Linear Systems”, J. Sci. Comput., 9, (1994).Google Scholar
- T.E. Booth, “Exponential Convergence for Monte Carlo Particle Transport”, Trans. Amer. Nucl. Soc., 50, 1986.Google Scholar
- J. Spanier, “A New Multistage Procedure for Systematic Variance Reduction in Monte Carlo”, SIAM J. Numer. Anal. 8, (1971).Google Scholar
- L. Li and J. Spanier, “Approximation of Transport Equations by Matrix Equations and Sequential Sampling”, in preparation.Google Scholar
- G.E. Forsythe and R.A. Leibler, “Matrix Inversion by a Monte Carlo Method”, Math. Tables Aids to Comp., 4 (1950).Google Scholar
- W. Wasow, “A Note on the Inversion of Matrices by Random Walks”, Math. Tables Aids to Comp., 6 (1952).Google Scholar
- J. Spanier, “A New Family of Estimators for Random Walk Problems”, J. Inst. Maths. Applics., 23 (1979).Google Scholar