Fast Multigrid Solvers for Transport with Forward-Peaked Scattering

本提案的主要目标是开发快速、高效和鲁棒的基于多网格的解算器，用于求解高度前峰散射下的线性玻尔兹曼输运方程。最近，该项目团队为一个模型问题开发了一种角度多重网格算法，该算法捕捉了二维“平面”模型中散射的玻尔兹曼输运方程的基本特征。因此，本项目的研究目标是将该方法扩展为一种高效有效的方法，用于真实介质中具有精确离散化的真正三维散射。将已经开发的技术扩展到三维的挑战包括改进离散化以允许不连续系数，以及在感兴趣的区域需要的局部网格细化。此外，项目团队将研究这些算法在二维和三维空间上的收敛性的理论分析，为这些算法的实际散射核的设计提供关键的见解。准确和高效的前峰散射模型在生物医学和核工程应用中都具有重要的意义。这一体系描述了用于治疗某些癌症肿瘤的放射治疗的电子束的散射，以及反应堆物理学和天体物理学中带电粒子的传输。尽管人们对高效、准确的模拟带电粒子输运的算法一直很感兴趣，但这个问题仍然面临着巨大的挑战。本文所考虑的方法为该领域的研究提供了一个新的方向。提出的工作直接导致生物医学和核工程师和科学家的模拟工具。PI和co-PI在计算科学和数学生物学社区中的积极作用确保了结果算法的及时和广泛传播。此外，该项目直接涉及一名博士生，他受到PI和共同PI的积极指导，有助于在一个重要的研究领域培养一名早期职业科学家。

The primary goal of this proposal is to develop fast, efficient, and robust multigrid-based solvers for the solution of the linear Boltzmann-transport equation in the regime of highly forward-peaked scattering. Recently, the project team have developed an angular multigrid algorithm for a model problem that captures the essential features of the Boltzmann-transport equation for scattering in a two-dimensional "Flatland" model. The research goals of this project are, thus, to extend this approach to an efficient and effective method for true three-dimensional scattering, in realistic media, with accurate discretizations. Among the challenges of extending the already developed technique to three dimensions are improving the discretization to allow discontinuous coefficients, and local grid refinement needed in regions of interest. Additionally, the project team will investigate theoretical analysis of the convergence of these algorithms, in both two and three spatial dimensions, providing critical insight into the design of these algorithms for realistic scattering kernels.Accurate and efficient models of forward-peaked scattering are of significant interest in both biomedical and nuclear engineering applications. This regime describes the scattering of electron beams, used in radiation therapy for the treatment of certain cancerous tumors, as well as the transport of charged particles in reactor physics and astrophysics. While there is a long history of interest in efficient and accurate algorithms for modeling charged-particle transport, the problem still poses formidable challenges. The approach considered here offers a new direction for research in this area. The proposed work leads directly to simulation tools for biomedical and nuclear engineers and scientists. The active roles of the PI and co-PI in the computational science and mathematical biology communities ensure timely and widespread dissemination of the resulting algorithms. Furthermore, the project directly involves a doctoral student, who is actively mentored by the PI and co-PI, contributing to the training of an early career scientist in an important field of research.