[论文解读] 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 problemsMathematics参考文献 19被引用 20

一句话总结

本文提出了一种自适应嵌套源项迭代（ASTI）方法，用于在L²范数下求解辐射转移方程，并提供经过认证的后验误差界。通过结合稳定化Galerkin变分格式、用于严格误差控制的不连续Galerkin（DPG）格式，以及基于误差界的自适应网格技术，该方法在保证计算精度的同时，通过低秩散射算子近似和矩阵压缩技术降低了计算成本。

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.

研究动机与目标

• 开发一种数值稳定、自适应的辐射转移方程求解算法，具备严格认证的误差控制能力。
• 解决在输运主导区域中，动力学模型缺乏误差控制求解器的问题。
• 通过自适应空间网格和散射算子的低秩近似，降低计算复杂度。
• 在动态调整的精度容差下，确保迭代格式的收敛性。
• 通过后验误差界，为辐射转移问题中的不确定性量化提供框架。

提出的方法

• 采用稳定的Petrov-Galerkin变分格式，求解参数化输运与辐射转移方程。
• 在函数空间中设计无限维迭代格式，每步保证固定误差减少量。
• 在由后验DPG误差界导出的动态更新精度容差内，对迭代过程进行近似。
• 利用后验DPG误差界指导自适应、参数相关的空间网格加密。
• 应用低秩与矩阵压缩技术，加速全局散射算子的计算。
• 使用自适应DPG离散化，在所需精度水平下求解参数相关线性输运方程。

实验结果

研究问题

• RQ1 是否可以在无限维函数空间中构造一种嵌套源项迭代方法，实现保证收敛与误差减少？
• RQ2 如何利用DPG格式的后验误差界，控制自适应迭代求解器中的精度？
• RQ3 基于误差界的自适应网格化对辐射转移问题的计算效率有何影响？
• RQ4 如何在保持误差控制的前提下，高效应用全局散射算子？
• RQ5 该方法是否能在比标准方法更少自由度的情况下实现经认证的精度？

主要发现

• ASTI算法在第10次迭代时达到最终误差0.00400132，自由度为42,179,602，表明其收敛至目标精度。
• 通过后验DPG误差界实现了经认证的误差控制，确保全局误差界始终高于实际误差。
• 基于后验误差估计的自适应空间网格显著降低了整体计算成本，优于均匀加密。
• 通过低秩近似高效应用了散射算子，实现了快速矩阵压缩并减少了求解时间。
• 收敛历史表明，内部误差容差比全局误差界更严格，表明误差管理有效。
• 最终解在不同角度方向上的可视化结果未出现非物理解振荡，证实了该方法的鲁棒性，且无需结构保持的修改。