On a Monte Carlo method for neutron transport criticality computations

Abstract

We give a stochastic representation of the principal eigenvalue of some homogeneous neutron transport operators. Our construction is based upon the Feynman-Kac formula for integral transport equations, and uses probabilistic techniques only. We develop a Monte Carlo method for criticality computations. We numerically test this method on various homogeneous and inhomogeneous problems, and compare our results with those obtained by standard methods.

Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics

“…For more details about Monte Carlo simulations of neutron transport processes, we refer to [13,19,20]. We denote byū NTJ h (x 0 ) the expectation of the score obtained by the NTJ algorithm when the initial position of the particle is x 0 .…”

“…In the case of all the algorithms described in Section 3, we actually have 20) where algo ∈ {SNJ, ANJ(α), UANJ(α), OANJ, NTJ} for any α > 0.…”

Simulating diffusions with piecewise constant coefficients using a kinetic approximation

“…First we make a global approximation of L by combining an exact simulation (See [32,33,41]) and a Romberg acceleration procedure. Then, we use only the approximation by the transport operator when one hits the surface of discontinuity of a and efficient simulations of the Brownian motion elsewhere.…”

Computing the principal eigenelements of some linear operators using a branching Monte Carlo method

“…1. A first idea is to estimate F N ðtÞ at two times t 0 and t 1 , and use the difference of the values at these times to estimate k 1 [28,32]. Another possible approach, developed in [25], is to find a time window ½t 0 ; t 1 , in which F N ðtÞ is a good approximation of FðtÞ.…”

“…The purpose of this article is to improve the results of [32] and [25]. To achieve this improvement, we propose a variance reduction scheme for the empirical approximation of FðtÞ which is very easy to implement.…”