Random Source Iteration Method: Mitigating the Ray Effect in the Discrete Ordinates Method
Jingyi Fu, Lei Li, Min Tang
TL;DR
This work addresses ray effects in the discrete ordinates method (DOM) for the steady-state radiative transport equation (RTE) by introducing Random Source Iteration (RSI), a randomized variant of the standard SI. RSI partitions the angular set $V=\{1,\dots,M\}$ into groups and samples one direction per group per iteration with weights $1/p_m^{(n)}$ to preserve unbiasedness, enabling parallel angular processing and reduced per-sample cost. The authors prove RSI is unbiased with respect to SI, its variance is uniformly bounded across iterations, and the RSI process is ergodic; the average over samples converges to the SI solution with a sampling convergence rate of $1/2$. Numerical experiments demonstrate that RSI mitigates ray effects without increasing overall computational cost and can exploit angular parallelism to achieve comparable accuracy to SI while handling larger angular sets.
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.