Random Source Iteration Method: Mitigating the Ray Effect in the Discrete Ordinates Method

TL;DR

This paper introduces a novel random source iteration (RSI) method for the discrete ordinates method (DOM) in radiative transport simulation, effectively reducing the ray effect without extra computational costs.

Contribution

The paper proposes a new RSI method that incorporates randomness into the SI process, providing unbiased results with bounded variance and proven ergodicity, to mitigate the ray effect in DOM.

Findings

RSI is unbiased and has bounded variance.

RSI convergence order is 1/2 with respect to sample number.

Numerical examples confirm RSI's effectiveness in reducing ray effects.

Abstract

The commonly used velocity discretization for simulating the radiative transport equation (RTE) is the discrete ordinates method (DOM). One of the long-standing drawbacks of DOM is the phenomenon known as the ray effect. Due to the high dimensionality of the RTE, DOM results in a large algebraic system to solve. The Source Iteration (SI) method is the most standard iterative method for solving this system. In this paper, by introducing randomness into the SI method, we propose a novel random source iteration (RSI) method that offers a new way to mitigate the ray effect without increasing the computational cost. We have rigorously proved that RSI is unbiased with respect to the SI method and that its variance is uniformly bounded across iteration steps; thus, the convergence order with respect to the number of samples is $1/2$. Furthermore, we prove that the RSI iteration process, as a Markov chain, is ergodic under mild assumptions. Numerical examples are presented to demonstrate the convergence of RSI and its effectiveness in mitigating the ray effect.