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On accelerated iterative schemes for anisotropic radiative transfer using residual minimization

Riccardo Bardin, Matthias Schlottbom

Abstract

We consider the iterative solution of anisotropic radiative transfer problems using residual minimization over suitable subspaces. We show convergence of the resulting iteration using Hilbert space norms, which allows us to obtain algorithms that are robust with respect to finite-dimensional realizations via Galerkin projections. We investigate in particular the behavior of the iterative scheme for discontinuous Galerkin discretizations in the angular variable in combination with subspaces that are derived from related diffusion problems. The performance of the resulting schemes is investigated in numerical examples for highly anisotropic scattering problems with heterogeneous parameters.

On accelerated iterative schemes for anisotropic radiative transfer using residual minimization

Abstract

We consider the iterative solution of anisotropic radiative transfer problems using residual minimization over suitable subspaces. We show convergence of the resulting iteration using Hilbert space norms, which allows us to obtain algorithms that are robust with respect to finite-dimensional realizations via Galerkin projections. We investigate in particular the behavior of the iterative scheme for discontinuous Galerkin discretizations in the angular variable in combination with subspaces that are derived from related diffusion problems. The performance of the resulting schemes is investigated in numerical examples for highly anisotropic scattering problems with heterogeneous parameters.
Paper Structure (24 sections, 8 theorems, 67 equations, 5 figures, 6 tables)

This paper contains 24 sections, 8 theorems, 67 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Approach and contribution
  3. Related works
  4. Outline
  5. Notation and preliminaries
  6. Function spaces
  7. Optical parameters and data
  8. Even-odd splitting
  9. Variational formulation
  10. Contraction properties of the source iteration
  11. Residual minimization
  12. Numerical realization
  13. Galerkin approximation
  14. Iterative scheme
  15. Formulation in terms of matrices
  16. ...and 9 more sections

Key Result

Lemma 1.1

\newlabellem:contractivity_residuals0 For $k\geq 0$ let $u_k,\ u_{k+1/2}$ be related by eq:SI2 and denote $r_k=\mathcal{R}(u_k)$ and $r_{k+1/2}=\mathcal{R}(u_{k+1/2})$ the respective preconditioned residuals defined in eq:preconditioned_residual. Then it holds that

Figures (5)

  • Figure 1: Left: geometry of the lattice problem in the checkerboard domain. White and gray areas are characterized by the optical parameters $\sigma_s=10$ and $\sigma_a=0.01$, while in the black zones $\sigma_s=0$ and $\sigma_a=1$. The internal source of radiation is $q=1$ in the gray square, $q=0$ outside of it. Right: Sketch of the spherical grid for the upper half sphere. \newlabelfig:checkerboard0
  • Figure 2: Residual decay for correction searched in the subspace $W_{h,N}^c$. From left to right, we plot ${\left\Vert \mathcal{R}_h(u_k) \right\Vert}_{\sigma_t}$; first row $g=0.1,\ 0.3,\ 0.5$; second row, $g=0.7,\ 0.9,\ 0.99$. The lines' style refers to number of even eigenfunctions of the scattering operator employed: $K=0$ (standard source iteration) densely dotted line; $K=1$ solid line; $K=6$ dotted line; $K=15$ dashed line. \newlabelfig:W1_residuals0
  • Figure 3: Residual decay for correction searched in the subspace $\stackon[.5pt]{W}{\sim} _{h,N}^c$. From left to right, we plot ${\left\Vert \mathcal{R}_h(u_k) \right\Vert}_{\sigma_t}$; first row $g=0.1,\ 0.3,\ 0.5$; second row, $g=0.7,\ 0.9,\ 0.99$. The lines style refers to the dimension of the corrections subspace: $K=1$ solid line; $K=6$ dotted line; $K=15$ dashed line. \newlabelfig:W1_tilde_residuals0
  • Figure 4: Residual decay for correction searched in the subspace $\stackon[.5pt]{W}{\sim} _{h,N}^{c,2}$. From left to right, we plot ${\left\Vert \mathcal{R}_h(u_k) \right\Vert}_{\sigma_t}$; first row $g=0.1,\ 0.3,\ 0.5$; second row, $g=0.7,\ 0.9,\ 0.99$. The lines' style refers to number of even eigenfunctions of the scattering operator employed: $K=1$ solid line; $K=6$ dotted line; $K=15$ dashed line. \newlabelfig:Wc2_residuals0
  • Figure 5: Residual decay for correction searched in the subspace $\stackon[.5pt]{W}{\sim} _{h,N}^{c,4}$. From left to right, we plot ${\left\Vert \mathcal{R}_h(u_k) \right\Vert}_{\sigma_t}$; first row $g=0.1,\ 0.3,\ 0.5$; second row, $g=0.7,\ 0.9,\ 0.99$. The lines' style refers to number of even eigenfunctions of the scattering operator employed: $K=1$ solid line; $K=6$ dotted line; $K=15$ dashed line. \newlabelfig:Wc4_residuals0

Theorems & Definitions (16)

  • Lemma 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proof 1
  • Lemma 3.1
  • Proof 2
  • Lemma 3.2
  • Proof 3
  • Lemma 3.3
  • Proof 4
  • ...and 6 more