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Analysis of eigenvalue clustering leads to optimal scaling in numerical radiative transfer

Pietro Benedusi, Simone Riva, Luca Belluzzi, Stefano Serra-Capizzano

Abstract

We consider a multidimensional polychromatic radiative transfer (RT) problem, accounting for scattering processes in a general form, i.e. anisotropic (dipole) scattering with partial frequency redistribution. Given a discrete ordinates discretization, we report the corresponding matrix structures, depending on model and discretization parameters. Despite the possibly dense nature of these matrices, the use of Krylov methods is effective (especially in the matrix-free context) and robust. We propose a theoretical analysis, using the spectral tools of the symbol theory, explaining why Krylov convergence is robust w.r.t. all the discretization parameters, even in the unpreconditioned case. In fact, the compactness of the continuous operators used in the modeling leads to zero-clustered dense matrix sequences plus identity, so that the clustering at the unity of the spectra is deduced. Numerical experiments confirm the theoretical results, which have a direct application, for example, in the simulation of radiative transfer in stellar atmospheres, a key problem in astrophysical research. In general, we demonstrate that optimal scaling with respect to RT discretization parameters is expected for Krylov solution strategies.

Analysis of eigenvalue clustering leads to optimal scaling in numerical radiative transfer

Abstract

We consider a multidimensional polychromatic radiative transfer (RT) problem, accounting for scattering processes in a general form, i.e. anisotropic (dipole) scattering with partial frequency redistribution. Given a discrete ordinates discretization, we report the corresponding matrix structures, depending on model and discretization parameters. Despite the possibly dense nature of these matrices, the use of Krylov methods is effective (especially in the matrix-free context) and robust. We propose a theoretical analysis, using the spectral tools of the symbol theory, explaining why Krylov convergence is robust w.r.t. all the discretization parameters, even in the unpreconditioned case. In fact, the compactness of the continuous operators used in the modeling leads to zero-clustered dense matrix sequences plus identity, so that the clustering at the unity of the spectra is deduced. Numerical experiments confirm the theoretical results, which have a direct application, for example, in the simulation of radiative transfer in stellar atmospheres, a key problem in astrophysical research. In general, we demonstrate that optimal scaling with respect to RT discretization parameters is expected for Krylov solution strategies.
Paper Structure (17 sections, 13 theorems, 76 equations, 9 figures, 3 tables)

This paper contains 17 sections, 13 theorems, 76 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Discretization and operator assembly
  4. Plane-parallel discretization
  5. Transfer operator $\Lambda_N$
  6. Scattering operator $\Sigma_N$
  7. Multi-dimensional case
  8. Global operator
  9. Spectral analysis
  10. Preliminaries
  11. Theoretical results
  12. Numerical experiments
  13. Monochromatic problem
  14. Isotropic problem
  15. Realistic 3D problem
  16. ...and 2 more sections

Key Result

Proposition 4.1

Suppose a sequence of matrices $\{A_N\}$, $A_N$ of size $d_N$ ($d_k<d_{k+1}$ for each $k\in \mathbb{N}$), is given, and $\{\{B_{N,m}\}:\, m\in\mathbb{N}^+\}$, $B_{N,m}$ of size $d_N$, is an a.c.s. for $\{A_N\}$ in the sense of Definition appr:seq. Assume that $\{B_{N,m}\}\sim_\sigma (h_m,K)$ and tha Furthermore assume that all $B_{N,m}, N_{N,m}, R_{N,m}$ are Hermitian and suppose that $\{B_{n,m}\}

Figures (9)

  • Figure 1: Representation of interpolation operators between two discretizations of a square domain $D$ in 2D. In the left ones, (used to compute transfer), give a ray $\bm{r}_k$, we consider $L_k=5$ rays, where the longest one ($\ell=1$) is partitioned with $M(k,1)=10$ nodes. On the right, we show the cartesian grid, where scattering is computed.
  • Figure 1: Percentage of eigenvalues with absolute value in the interval [0.999,1] for the monochromatic problem.
  • Figure 2: Monochromatic solution of \ref{['eq:mono_sol']} discretized with $N_s=200$ and $N_\Omega=N_r=12$, with a discontinuity for $\mu=0$.
  • Figure 2: Percentage of eigenvalues with absolute value in the interval [0.999,1] using $N_\mu=12$.
  • Figure 3: GMRES and BiCGStab convergence for different discretization parameters for the monochromatic problem.
  • ...and 4 more figures

Theorems & Definitions (23)

  • Definition 4.1
  • Definition 4.2
  • Proposition 4.1
  • Definition 4.3: sparsely unbounded matrix sequence
  • Definition 4.4: sparsely vanishing matrix sequence
  • Proposition 4.2
  • Definition 4.5
  • Remark 4.1
  • Theorem 4.1
  • Theorem 4.2
  • ...and 13 more