IJPAM: Volume 59, No. 4 (2010)

GEOMETRIC CONVERGENCE OF SECOND GENERATION
ADAPTIVE MONTE CARLO ALGORITHMS FOR
GENERAL TRANSPORT PROBLEMS BASED
ON CORRELATED SAMPLING

Rong Kong$^1$, Jerome Spanier$^2$
$^1$Claremont Graduate University
150 E. 10-th St., Claremont, California, 91711, USA
e-mail: kongr413@yahoo.com
$^2$Beckman Laser Institute and Medical Clinic
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