A flexible GMRES solver with reduced order model enhanced synthetic acceleration preconditioner for parametric radiative transfer equation

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.