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A Flexible GMRES Solver with Reduced Order Model Enhanced Synthetic Acceleration Preconditioenr for Parametric Radiative Transfer Equation

Zhichao Peng

TL;DR

The paper extends ROMSAD from SI to a Krylov setting by embedding a ROM-based synthetic acceleration into Flexible GMRES, enabling efficient solutions of parametric RTEs. It reformulates the ideal kinetic correction to avoid direct solves within the Krylov framework and uses a greedy offline algorithm to build a compact ROM that captures parameter-induced low-rank structure. Numerical experiments across lattice, pin-cell, and scattering-variation problems show that FG-MRES-ROMSAD can substantially outperform GMRES-DSA and SI-DSA, with robustness maintained even when the ROM is relatively approximate. The approach offers significant offline-online efficiency gains and suggests a viable pathway for multi-query RTE applications in uncertainty quantification and inverse problems.

Abstract

Parametric radiative transfer equation (RTE) occurs in multi-query applications such as uncertainty quantification, inverse problems, and sensitivity analysis, which require solving RTE multiple times for a range of parameters. Consequently, efficient iterative solvers are highly desired. Classical Synthetic Acceleration (SA) preconditioners for RTE build on low order approximations to an ideal kinetic correction equation such as its diffusion limit in Diffusion Synthetic Acceleration (DSA). Their performance depends on the effectiveness of the underlying low order approximation. In addition, they do not leverage low rank structures with respect to the parameters of the parametric problem. To address these issues, we proposed a ROM-enhanced SA strategy, called ROMSAD, under the Source Iteration framework in Peng (2024). In this paper, we further extend the ROMSAD preconditioner to flexible general minimal residual method (FGMRES). The main new advancement is twofold. First, after identifying the ideal kinetic correction equation within the FGMRES framework, we reformulate it into an equivalent form, allowing us to develop an iterative procedure to construct a ROM for this ideal correction equation without directly solving it. Second, we introduce a greedy algorithm to build the underlying ROM for the ROMSAD preconditioner more efficiently. Our numerical examples demonstrate that FGMRES with the ROMSAD preconditioner (FGMRES-ROMSAD) is more efficient than GMRES with the right DSA preconditioner. Furthermore, when the underlying ROM in ROMSAD is not highly accurate, FGMRES-ROMSAD exhibits greater robustness compared to Source Iteration accelerated by ROMSAD.

A Flexible GMRES Solver with Reduced Order Model Enhanced Synthetic Acceleration Preconditioenr for Parametric Radiative Transfer Equation

TL;DR

The paper extends ROMSAD from SI to a Krylov setting by embedding a ROM-based synthetic acceleration into Flexible GMRES, enabling efficient solutions of parametric RTEs. It reformulates the ideal kinetic correction to avoid direct solves within the Krylov framework and uses a greedy offline algorithm to build a compact ROM that captures parameter-induced low-rank structure. Numerical experiments across lattice, pin-cell, and scattering-variation problems show that FG-MRES-ROMSAD can substantially outperform GMRES-DSA and SI-DSA, with robustness maintained even when the ROM is relatively approximate. The approach offers significant offline-online efficiency gains and suggests a viable pathway for multi-query RTE applications in uncertainty quantification and inverse problems.

Abstract

Parametric radiative transfer equation (RTE) occurs in multi-query applications such as uncertainty quantification, inverse problems, and sensitivity analysis, which require solving RTE multiple times for a range of parameters. Consequently, efficient iterative solvers are highly desired. Classical Synthetic Acceleration (SA) preconditioners for RTE build on low order approximations to an ideal kinetic correction equation such as its diffusion limit in Diffusion Synthetic Acceleration (DSA). Their performance depends on the effectiveness of the underlying low order approximation. In addition, they do not leverage low rank structures with respect to the parameters of the parametric problem. To address these issues, we proposed a ROM-enhanced SA strategy, called ROMSAD, under the Source Iteration framework in Peng (2024). In this paper, we further extend the ROMSAD preconditioner to flexible general minimal residual method (FGMRES). The main new advancement is twofold. First, after identifying the ideal kinetic correction equation within the FGMRES framework, we reformulate it into an equivalent form, allowing us to develop an iterative procedure to construct a ROM for this ideal correction equation without directly solving it. Second, we introduce a greedy algorithm to build the underlying ROM for the ROMSAD preconditioner more efficiently. Our numerical examples demonstrate that FGMRES with the ROMSAD preconditioner (FGMRES-ROMSAD) is more efficient than GMRES with the right DSA preconditioner. Furthermore, when the underlying ROM in ROMSAD is not highly accurate, FGMRES-ROMSAD exhibits greater robustness compared to Source Iteration accelerated by ROMSAD.

Paper Structure

This paper contains 24 sections, 55 equations, 5 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model equation and full order discretization
  3. Discrete ordinates ($S_N$) angular discretization
  4. Upwind discontinuous Galerkin spatial discretization
  5. Matrix-vector formulation
  6. Krylov Solver and Source Iteration with Classical Synthetic Acceleration
  7. Source Iteration and Synthetic Acceleration
  8. GMRES with SA based preconditioner
  9. Limitations of DSA
  10. Reduced order model enhanced preconditioner for Krylov method
  11. ROM for affine parametric problem
  12. Flexible GMRES
  13. ROM-enhanced SA preconditioner
  14. ROM construction
  15. Ideal correction under the Krylov framework and data generation
  16. ...and 9 more sections

Figures (5)

  • Figure 1: The set-up and a reference solution for the lattice problem in Sec. \ref{['sec:lattice']}. Left: the set-up for the lattice problem. Black: pure absorption regions with $(\sigma_a,\sigma_s)=(\mu_a,0)$. White and orange: pure scattering regions with $(\sigma_a,\sigma_s)=(0,\mu_s)$. Orange: constant source term with $G=1$. Right: the reference solution under log-scale for $(\mu_a,\mu_s)=(100,1.0)$. We want to point out that negative scalar flux can be generated in this example, but it will not break the linear solver, since we are not considering thermal radiation here. When generating the plot under log-scale, we take $\max(10^{-16},\phi(\mathbf{x}))$.
  • Figure 2: The set-up and reference solutions for the pin-cell problem in Sec. \ref{['sec:pin-cell']}. Left: problem set-up. Right: $\phi$ for $(\mu_s,\mu_a)=(0.065886,0.11397)$ (under log-scale).
  • Figure 3: Convergence history for FGMRES-ROMSAD and SI-ROMSAD with $\mathfrak{w}=1$ and $\epsilon_{\textrm{ROM}}=10^{-7}$ for the pin-cell problem in Sec. \ref{['sec:pin-cell']} with $(\mu_s,\mu_a)=(0.065886,0.11397)$ (under log-scale).
  • Figure 4: Left: $\sigma_s(\mathbf{x})$ with $\mu_s=99.9$ for the variable scattering problem in Sec. \ref{['sec:variable-scattering']}. Right: corresponding angular flux.
  • Figure 5: Reference solution with $(\mu_s,\mu_{\textrm{bc}})=(3,1)$ for the two-material problem with parametric inflow boundary conditions in Sec. \ref{['sec:parametric-bc']}.

Theorems & Definitions (4)

  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 5.1