[Paper Review] An Adaptive Nested Source Term Iteration for Radiative Transfer Equations

Wolfgang Dahmen, Felix Gruber+1 more|arXiv (Cornell University)|Oct 16, 2018

Numerical methods in inverse problems19 references20 citations

TL;DR

This paper introduces an Adaptive Nested Source Term Iteration (ASTI) method for solving radiative transfer equations with certified a posteriori error bounds in the L2 norm. By combining stabilized Petrov-Galerkin variational formulations, Discontinuous Petrov-Galerkin (DPG) schemes for rigorous error control, and adaptive meshing driven by error bounds, the method enables convergence with guaranteed accuracy while reducing computational cost through low-rank scattering operator approximation and matrix compression.

ABSTRACT

We propose a new approach to the numerical solution of radiative transfer equations with certified a posteriori error bounds. A key role is played by stable Petrov--Galerkin type variational formulations of parametric transport equations and corresponding radiative transfer equations. This allows us to formulate an iteration in a suitable, infinite dimensional function space that is guaranteed to converge with a fixed error reduction per step. The numerical scheme is then based on approximately realizing this iteration within dynamically updated accuracy tolerances that still ensure convergence to the exact solution. To advance this iteration two operations need to be performed within suitably tightened accuracy tolerances. First, the global scattering operator needs to be approximately applied to the current iterate within a tolerance comparable to the current accuracy level. Second, parameter dependent linear transport equations need to be solved, again at the required accuracy of the iteration. To ensure that the stage dependent error tolerances are met, one has to employ rigorous a posteriori error bounds which, in our case, rest on a Discontinuous Petrov--Galerkin (DPG) scheme. These a posteriori bounds are not only crucial for guaranteeing the convergence of the perturbed iteration but are also used to generate adapted parameter dependent spatial meshes. This turns out to significantly reduce overall computational complexity. Since the global operator is only applied, we avoid the need to solve linear systems with densely populated matrices. Moreover, the approximate application of the global scatterer accelerated through low-rank approximation and matrix compression techniques. The theoretical findings are illustrated and complemented by numerical experiments with non-trivial scattering kernels.

Motivation & Objective

• To develop a numerically stable, adaptive algorithm for radiative transfer equations with rigorously certified error control.
• To address the lack of error-controlled solvers for kinetic models in transport-dominated regimes.
• To reduce computational complexity through adaptive spatial meshes and low-rank approximation of the scattering operator.
• To ensure convergence of the iterative scheme under dynamically adjusted accuracy tolerances.
• To provide a framework for uncertainty quantification in radiative transfer problems via a posteriori error bounds.

Proposed method

• Employs a stable Petrov-Galerkin variational formulation for parametric transport and radiative transfer equations.
• Designs an infinite-dimensional iterative scheme in function space with guaranteed fixed error reduction per step.
• Approximates the iteration within dynamically updated accuracy tolerances derived from a posteriori DPG error bounds.
• Uses a posteriori DPG bounds to guide adaptive, parameter-dependent spatial mesh refinement.
• Applies low-rank and matrix compression techniques to accelerate the global scattering operator application.
• Solves parameter-dependent linear transport equations at required accuracy levels using adaptive DPG discretizations.

Experimental results

Research questions

• RQ1 Can a nested source term iteration be formulated in infinite-dimensional function space with guaranteed convergence and error reduction?
• RQ2 How can a posteriori error bounds from DPG schemes be used to control accuracy in an adaptive, iterative solver?
• RQ3 What is the impact of adaptive meshing driven by error bounds on computational efficiency in radiative transfer?
• RQ4 How can the global scattering operator be applied efficiently while maintaining error control?
• RQ5 Can the method achieve certified accuracy with reduced degrees of freedom compared to standard approaches?

Key findings

• The ASTI algorithm achieved a final error of 0.00400132 at iteration 10 with 42,179,602 degrees of freedom, demonstrating convergence to the target accuracy.
• The method achieved certified error control via a posteriori DPG bounds, ensuring the global error bound was always above the actual error.
• Adaptive spatial meshes based on a posteriori error estimates significantly reduced overall computational cost compared to uniform refinement.
• The scattering operator was efficiently applied using low-rank approximation, enabling fast matrix compression and reduced solution time.
• The convergence history showed that interior error tolerances were tighter than the global error bound, indicating effective error management.
• The final solution, visualized across angular directions, showed no unphysical oscillations, confirming the method's robustness without structure-preserving modifications.