Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence Analysis of the Random Ordinate Method for Mitigating the Ray Effect

Lei Li, Min Tang, Yuqi Yang

Abstract

The Discrete Ordinates Method (DOM) is widely used for velocity discretization in radiative transport simulations. However, DOM tends to exhibit the ray effect when the velocity discretization is not sufficiently refined, a limitation that is well documented. To counter this, we have developed the Random Ordinates Method (ROM) by integrating randomness into the velocity discretization, which mitigates the ray effect without incurring additional computational costs. ROM partitions the velocity space into n cells, selects a random ordinate from each cell, and solves a DOM system with these ordinates. It leverages the average of multiple samples to achieve a higher convergence order, especially for solutions with low regularity in the velocity variable. In this work, we provide a detailed convergence analysis for ROM, focusing on bias and single-run errors. This analysis is crucial for determining the necessary mesh size and the optimal number of samples required to attain a specified level of accuracy.

Convergence Analysis of the Random Ordinate Method for Mitigating the Ray Effect

Abstract

The Discrete Ordinates Method (DOM) is widely used for velocity discretization in radiative transport simulations. However, DOM tends to exhibit the ray effect when the velocity discretization is not sufficiently refined, a limitation that is well documented. To counter this, we have developed the Random Ordinates Method (ROM) by integrating randomness into the velocity discretization, which mitigates the ray effect without incurring additional computational costs. ROM partitions the velocity space into n cells, selects a random ordinate from each cell, and solves a DOM system with these ordinates. It leverages the average of multiple samples to achieve a higher convergence order, especially for solutions with low regularity in the velocity variable. In this work, we provide a detailed convergence analysis for ROM, focusing on bias and single-run errors. This analysis is crucial for determining the necessary mesh size and the optimal number of samples required to attain a specified level of accuracy.

Paper Structure

This paper contains 12 sections, 12 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. The model and problem setup
  3. The regularized model in the slab geometry and ROM
  4. Expansion of the solution
  5. The transport operator $\mathcal{A}_{\mu}$
  6. The iteration operator $\mathcal{T}$
  7. Expansion in operator form
  8. Main result and the proof
  9. Some important estimates
  10. Discussion
  11. An estimate of the error introduced by regularization
  12. Detailed Derivation from 3D Radiative Transfer Equation to Slab Geometry Equation

Key Result

Theorem 3.1

\newlabelthm:errortransport0 Consider the regularized model for eq:1D_analysisequation and suppose that the rescaled weights $\alpha_{\ell}$ are uniformly bounded. Then, there exists $n_0>0$ such that for $n>n_0$, the expected single run error satisfies and the bias satisfies Here, the norm used is the $L^2(\sigma_T)$ norm in eq:L2norm.

Theorems & Definitions (21)

  • Theorem 3.1: main result
  • Proof 1: Proof of Theorem \ref{['thm:errortransport']}
  • Lemma 4.1
  • Proof 2
  • Corollary 4.2
  • Proof 3
  • Lemma 4.3
  • Lemma 4.4
  • Proposition 4.5
  • Proof 4
  • ...and 11 more