Response Matrix Method–Based Importance Solver and Variance Reduction Scheme in the Serpent 2 Monte Carlo Code

Abstract

A deterministic importance solver has been implemented as an internal subroutine in the Serpent 2 Monte Carlo code for the purpose of producing weight-window meshes for variance reduction. The routine solves the adjoint transport problem using the response matrix method with coupling coefficients obtained from a conventional forward Monte Carlo simulation. The methodology can be applied to photon and neutron external source problems, and the solver supports multiple energy groups and several mesh types. Importances can be generated with respect to multiple responses, and an iterative global variance reduction sequence enables distributing the transported particle population evenly throughout the geometry. This paper describes the methodology applied in the response matrix solver and presents a verification for the generated importance functions through simple demonstrations. A practical example involving a photon shielding problem is included for performance evaluation.

Keywords:

• Serpent 2
• Monte Carlo
• variance reduction
• weight windows
• response matrix method

Notes

^a The hat symbol is used throughout this paper to distinguish global (geometry-wide) quantities from local (intracell) parameters.

^b Normalizing the distributions according to the highest source importance eliminates splitting altogether, but may in the case of steep importance profiles result in very low sampling efficiency. The normalization in Serpent 2 is currently performed in such a way that 80% of source particles are emitted in cells where $I_{\mathrm{s}}<1$. This increases the efficiency by allowing some splitting to occur. The selection is somewhat arbitrary, and the optimization of the source routine was still under way at the time of this writing. A widely used solution is to apply the Consistent Adjoint-Driven Importance Sampling (CADIS) method, which ensures that all particles are emitted within the weight-window boundaries.¹⁰

^c Strictly speaking, this only applies to a case where weight-window boundaries are set extremely narrow and the particle weights are constrained tightly around the inverse importance.

^d Consider, for example, a system formed by source and two detectors (a) and (b), positioned in such way that all particles contributing to detector (b) also pass through detector (a). When the variance reduction scheme pushes the particles toward (b), the number of scores for (a) is also increased, regardless of how the importances are scaled with respect to each other.

^e To give a practical example, the memory footprint of a 100 $\times$ 100 $\times$ 100 mesh with nine energy groups is around 80 Gbyte.

^f This example also demonstrates an interesting outcome regarding the number of scores. As expected, the mean values are approximately the same, but the statistical errors differ by a factor of 5. This implies that the statistical errors of tallies obtained with a weight window–based variance reduction scheme may differ, even though they receive the same average number of scores.