IJPAM: Volume 59, No. 4 (2010)
ADAPTIVE MONTE CARLO ALGORITHMS FOR
GENERAL TRANSPORT PROBLEMS BASED
ON CORRELATED SAMPLING



150 E. 10-th St., Claremont, California, 91711, USA
e-mail: kongr413@yahoo.com

University of California
1002, Health Science Road E., Irvine, California 92612, USA
e-mail: jspanier@uci.edu
Abstract.If standard variance reduction methods are applied sequentially in a Monte
Carlo simulation, it is possible to achieve geometric reduction of the
variance (with probability one) to arbitrarily low error levels. We
previously applied this idea to the solution of general radiation transport
equations (RTE) by estimating a finite number of coefficients of expansion
of the solution in a fixed set of basis functions. However, this first
generation adaptive method proves to be impractical because of growth in the
number of expansion coefficients and because of the accumulation of
computational errors needed in the general case. If the goal of arbitrary accuracy everywhere in phase space is replaced by the
more practical goal of estimating a small number of linear
functionals of the RTE solution (the measurements, or observables of the
system), real gains in computational efficiency can be obtained. We recently
introduced a second generation adaptive algorithm based on this principle
and illustrated its rapid convergence and increased computational efficiency
when compared with conventional and first generation Monte Carlo algorithms.
In this paper we establish rigorously the geometric convergence of the new
algorithm and exhibit its convergence characteristics when solving two
dimensional transport problems The new algorithm shows promise of making
possible the efficient solution of many continuous transport problems not
amenable to either conventional or earlier adaptive methods.
Received: February 2, 2010
AMS Subject Classification: 82D75
Key Words and Phrases: transport equation, geometrically convergent Monte Carlo algorithms
Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 59
Issue: 4