Geometrically Convergent Learning Algorithms for Global Solutions of Transport Problems

Abstract

In 1996 Los Alamos National Laboratory initiated an ambitious five year research program aimed at achieving geometric convergence for Monte Carlo solutions of difficult neutron and photon transport problems. Claremont students, working with the author in Mathematics Clinic projects that same year and subsequently, have been partners in this undertaking. This paper summarizes progress made at Claremont over the two year period, with emphasis on recent advances.

The Claremont approach has been to maintain as much generality as possible, aiming ultimately at the Monte Carlo solution of quite general transport equations while using various model transport problems — both discrete and continuous — to establish feasibility. As far as we are aware, prior to this effort, only the discrete case had been seriously attacked by sequential sampling methods: by Halton beginning in 1962 [1] and subsequently by Kollman in his 1993 Stanford dissertation [2]. In work performed in Claremont, an adaptive importance sampling algorithm consistently outperformed a sequential correlated sampling algorithm based on Halton’s ideas for matrix problems. These findings are contrary to what Halton reported in 1962 and in subsequent papers.

These learning algorithms based on very different Monte Carlo strategies have recently been successfully extended to continuous problems. This paper outlines the methods and ideas employed, sketches the algorithms used and exhibits the geometric convergence obtained. A rationale for the results obtained so far and an indication of some of the remaining obstacles to achieving fully practical computation of global transport solutions by these means is also presented.