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

TL;DR

This paper introduces a novel ROM-enhanced synthetic acceleration preconditioner integrated with FGMRES for efficiently solving parametric radiative transfer equations, outperforming traditional methods in robustness and efficiency.

Contribution

It extends the ROMSAD preconditioner to FGMRES, developing an iterative ROM construction and a greedy algorithm for improved efficiency and robustness.

Findings

FGMRES-ROMSAD outperforms GMRES with DSA in efficiency.

FGMRES-ROMSAD is more robust when ROM accuracy is limited.

Numerical tests confirm the effectiveness of the proposed method.

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.