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Random ordinate method for mitigating the ray effect in radiative transport equation simulations

Lei Li, Min Tang, Yuqi Yang

TL;DR

This paper tackles the ray effect inherent in the discrete ordinates method (DOM) for solving the radiative transport equation (RTE) by introducing the Random Ordinate Method (ROM). ROM partitions the velocity space into cells and samples one random ordinate per cell, solving many independent DOM problems and averaging the results, which yields higher effective convergence in velocity space and mitigates ray effects without a substantial cost increase. The authors provide formal bias and mean-error analyses and demonstrate through slab and XY geometry experiments that ROM can outperform DOM in low-regularity velocity regimes, while remaining easy to implement and highly parallelizable. The results indicate that ROM achieves comparable overall computational costs to DOM and offers a robust, geometry-independent strategy to improve accuracy and suppress ray-induced artifacts in practical radiative transport simulations. The work suggests promising extensions to multiscale and diffusion-limit regimes and invites exploration of mass-conservation adjustments and importance-sampling variants.

Abstract

The Discrete Ordinates Method (DOM) is the most widely used velocity discretization method for simulating the radiative transport equation. However, the ray effect is a long-standing drawback of DOM. In benchmark tests that exhibit the ray effect, we observe low regularity in the velocity variable of the solution. To address this issue, we propose a Random Ordinate Method (ROM) to mitigate the ray effect. Compared to other strategies proposed in the literature for mitigating the ray effect, ROM offers several advantages: 1) For benchmark tests that exhibit ray effect, the computational cost is lower than that of the DOM; 2) it is simple and requires minimal changes to existing DOM-based code; 3) it is easily parallelizable and independent of the problem setup. A formal analysis is presented for the convergence orders of the error and bias. Numerical tests demonstrate the reduction in computational cost compared to DOM, as well as its effectiveness in mitigating the ray effect.

Random ordinate method for mitigating the ray effect in radiative transport equation simulations

TL;DR

This paper tackles the ray effect inherent in the discrete ordinates method (DOM) for solving the radiative transport equation (RTE) by introducing the Random Ordinate Method (ROM). ROM partitions the velocity space into cells and samples one random ordinate per cell, solving many independent DOM problems and averaging the results, which yields higher effective convergence in velocity space and mitigates ray effects without a substantial cost increase. The authors provide formal bias and mean-error analyses and demonstrate through slab and XY geometry experiments that ROM can outperform DOM in low-regularity velocity regimes, while remaining easy to implement and highly parallelizable. The results indicate that ROM achieves comparable overall computational costs to DOM and offers a robust, geometry-independent strategy to improve accuracy and suppress ray-induced artifacts in practical radiative transport simulations. The work suggests promising extensions to multiscale and diffusion-limit regimes and invites exploration of mass-conservation adjustments and importance-sampling variants.

Abstract

The Discrete Ordinates Method (DOM) is the most widely used velocity discretization method for simulating the radiative transport equation. However, the ray effect is a long-standing drawback of DOM. In benchmark tests that exhibit the ray effect, we observe low regularity in the velocity variable of the solution. To address this issue, we propose a Random Ordinate Method (ROM) to mitigate the ray effect. Compared to other strategies proposed in the literature for mitigating the ray effect, ROM offers several advantages: 1) For benchmark tests that exhibit ray effect, the computational cost is lower than that of the DOM; 2) it is simple and requires minimal changes to existing DOM-based code; 3) it is easily parallelizable and independent of the problem setup. A formal analysis is presented for the convergence orders of the error and bias. Numerical tests demonstrate the reduction in computational cost compared to DOM, as well as its effectiveness in mitigating the ray effect.
Paper Structure (19 sections, 63 equations, 15 figures, 3 tables)

This paper contains 19 sections, 63 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Ray effects and low regularity
  3. Random ordinate method
  4. Algorithmic details of ROM
  5. Formal analysis of the bias and mean error
  6. Comparison of the computational cost of DOM and ROM
  7. Numerical experiments
  8. The slab geometry
  9. Convergence with respect to the number of sampled simulations
  10. Convergence with respect to the mesh sizes
  11. Comparison of ROM with DOM
  12. Comparison of the complexity.
  13. The X-Y geometry case
  14. Convergence order
  15. Comparison of complexity
  16. ...and 4 more sections

Figures (15)

  • Figure 1: Demonstration of the ray effects. The average densities $\phi$ calculated with different numbers of ordinates are displayed. The numerical results are calculated with $100 \times 100$ spatial cells and uniform quadrature. (a)(b)(c): isotropic scattering kernel with $g=0$ in \ref{['eq:anisotropic kernel']}; (d)(e)(f) anisotropic scattering kernel with $g=0.9$ in \ref{['eq:anisotropic kernel']}.
  • Figure 1: Example \ref{['example:1Dthreecases']}: Convergence orders of DOM with different inflow boundary conditions in \ref{['eq:2.12']}-\ref{['eq:2.14']}. (a): Uniform quadratures of different sizes; (b): Gaussian quadratures of different sizes. Here $\Delta\mu=\frac{1}{M}$.
  • Figure 1: Schematic diagram of selected ordinates on the surface of a 3D unit sphere and their corresponding projection to the 2D unit disk. (a)(b) Uniform quadrature; (c)(d) Gaussian quadrature.
  • Figure 2: The convergence of ROM with respect to the number of sampled simulations $t$. (a): the MC error defined in \ref{['eq:1dMCerror']} for different cases in Example \ref{['example:1Dthreecases']}; (b): the MC error for different scattering kernel in Example \ref{['2D disk_ray effect']}. $t$ is the number of sampled simulations in ROM.
  • Figure 3: The convergence orders of ROM in velocity in slab geometry. Here $\Delta \mu=\frac{2}{n}$,$n=2,4,8,16$. (a): the errors defined in \ref{['eq:norm-error']} for different cases in \ref{['eq:2.12']}-\ref{['eq:2.14']}; (b): the MC error for different cases in \ref{['eq:2.12']}-\ref{['eq:2.14']}. $t$ is the number of sampled simulations in ROM. The values $t = 1280$ and $20480$ were selected. In practice, a magnitude of $10^3$ usually suffices, but here we need $t$ large enough for the MC error to reach the asymptotic, high-order regime so that the observed slope matches the theoretical value.
  • ...and 10 more figures

Theorems & Definitions (3)

  • Example 2.1
  • Remark 3.1
  • Example 4.1