### A Monte Carlo-Deterministic Method for Global Transport Calculations

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** Allan Wollaber,
University of Michigan **

Two fundamentally distinct classes of computational techniques have been developed for numerically solving particle transport problems: *stochastic* (Monte Carlo) methods, based on random simulations of a large number of individual particle histories, and *deterministic* methods, in which the linear Boltzmann equation is approximated by a large algebraic system of equations defined on a grid. Within the past decade, elements of Monte Carlo and deterministic methods have been combined to yield *hybrid* methods that can significantly improve upon the properties of purely Monte Carlo or deterministic methods. We here propose a new hybrid technique in which an inexpensive deterministic calculation facilitates the more efficient solution of a global Monte Carlo particle transport problem.

We specifically consider deep penetration problems, for which accurate estimates of the flux are sought everywhere within the physical system. Such problems are usually solved by deterministic methods, because they automatically generate solutions for all points on the space-angle-energy grid. Monte Carlo methods are infrequently used for global problems because of the very large statistical uncertainties that occur in “deep” subregions, located far from sources. The only common Monte Carlo method available for these problems is the nonanalog technique of absorption weighting. In this method, large numbers of Monte Carlo particles do reach the “deep” parts of the system, but the variance in their weights is generally large, causing large variances in estimates of the flux in these regions. The new hybrid technique remedies this by using a Monte Carlo method to calculate the multiplicative correction to an inexpensive deterministic approximation. The essential idea is that if the deterministic approximation is reasonably accurate, then the gradient in the multiplicative correction is smaller, allowing for less variance in particle weights and hence an improved overall solution. We demonstrate this effect by implementing the method for a simple problem.

**Abstract Author(s):** Allan Wollaber<br />Edward Larsen