A Stochastic Calculus Approach to Boltzmann Transport

TL;DR

This paper introduces a stochastic calculus framework linking particle paths in Monte Carlo transport methods to both flux and adjoint flux densities, providing a rigorous mathematical foundation for their simultaneous approximation.

Contribution

It establishes a rigorous stochastic calculus approach showing that particle paths in source iteration approximate both flux and adjoint flux densities simultaneously.

Findings

Particle paths relate to both flux and adjoint flux densities.

Stochastic differential equations represent particle paths and their adjoints.

Simulations confirm the dual approximation capability of particle paths.

Abstract

Traditional Monte Carlo methods for particle transport utilize source iteration to express the solution, the flux density, of the transport equation as a Neumann series. Our contribution is to show that the particle paths simulated within source iteration are associated with the adjoint flux density and the adjoint particle paths are associated with the flux density. We make our assertion rigorous through the use of stochastic calculus by representing the particle path used in source iteration as a solution to a stochastic differential equation (SDE). The solution to the adjoint Boltzmann equation is then expressed in terms of the same SDE and the solution to the Boltzmann equation is expressed in terms of the SDE associated with the adjoint particle process. An important consequence is that the particle paths used within source iteration simultaneously provide Monte Carlo approximations of the flux density and adjoint flux density. Monte Carlo simulations are presented to support the simultaneous use of the particle paths.