Preconditioning and Solver Optimization Ideas for Radiative Transfer

Abstract

In this paper, radiative transfer and time-dependent transport of radiation energy in participating media are modeled using a first-order spherical harmonics method (P1) and radiation diffusion. Partial differential equations for P1 and radiation diffusion are discretized by a variational form of the equations using support operators. Choices made in the discretization result in a symmetric positive definite (SPD) system of linear equations. Modeling multidimensional domains with complex geometries requires a very large system of linear equations with 10s of millions of elements. The computational domain is decomposed into a large number of subdomains that are solved on separate processors resulting in a massively parallel application. The linear system of equations is solved with a preconditioned conjugate gradient method. Various preconditioning techniques are compared in this study. Simple preconditioning techniques include: diagonal scaling, Symmetric Successive Over Relaxation (SSOR), and block Jacobi with SSOR as the block solver. Also, a two-grid multigrid-V-cycle method with aggressive coarsening is explored for use in the problems presented. Results show that depending on the test problem, simple preconditioners are effective, but the more complicated preconditioners such as an algebraic multigrid or the geometric multigrid are most efficient, particularly for larger problems and longer simulations. Optimal preconditioning varies depending on the problem and on how the physical processes evolve in time. For the insitu preconditioning techniques—SSOR and block Jacobi—a fuzzy controller can determine the optimal reconditioning process. Discussions of the current knowledge-based controller, an optimization search algorithm, are presented. Discussions of how this search algorithm can be incorporated into the development of data-driven controller incorporating clustering and subsequent construction of the fuzzy model from partitions are also discussed.

A Parallel First-Order Spherical Harmonics (P1) Matrix-Free Method for Radiative Transport

“…The method is commonly used to solve radiation transport and is described by Knoll et al [22] and Carrington [12]. The method proceeds by solving Eq.…”

“…The algebraic multigrid preconditioner has been shown to be very effective when the cost of constructing the system is amortized over many processors and when a large number of cells is involved [12]. Knoll and Keyes provide an overview of preconditioning techniques that are useful for the matrix-free method [24].…”

“…The linear equation solver, the Krylov iterative solver, utilizes a suite of preconditioners, an in-situ system described by Carrington and Mousseau in 2005 [12] or the Los Alamos Algebraic Multigrid (LAMG) method originally developed by Joubert [13]. The algebraic multigrid preconditioner has been shown to be very effective when the cost of constructing the system is amortized over many processors and when a large number of cells is involved [12].…”

“…Krishnamoorthy et al, discuss the process of a parallel implementation of the P1 method using a matrix formulation and various solution techniques [11]. Recently, Carrington and Mousseau demonstrated parallel solution methods developed for the P 1 (and diffusion) models for both 2-D and 3-D geometries [12]. These preconditioners include an algebraic multigrid method [13] and the preconditioners described here.…”

“…In this article, I demonstrate the efficiencies of parallel scaling for the matrix-free implementation along with the performance of the preconditioners, symmetric successive overrelaxation (SSOR) and block Jacobi. These preconditioners effectively reduce the amount of computation work required by the selected Krylov solver in both the matrix formulation [12] and with the matrix-free method.…”