A Reduced Basis Method for Radiative Transfer Equation
A Reduced Basis Method for Radiative Transfer Equation
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DOI：
10.1007/s10915-022-01782-2
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发表时间：
2021-03
影响因子：
2.5
通讯作者：
Zhichao Peng;Yanlai Chen;Yingda Cheng;Fengyan Li
中科院分区：
文献类型：
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作者：
Zhichao Peng;Yanlai Chen;Yingda Cheng;Fengyan Li
线性动力学输运方程在光学层析成像、辐射传递和中子输运中起着至关重要的作用。物理变量和速度/角度变量的高维性以及问题本质上是多尺度的，这是阻碍它们有效和准确地进行数值解析的根本困难。利用扩散极限暗示的隐藏低秩结构的存在性，在这项工作中，我们设计并测试了线性辐射传递方程的角空间降阶模型，这是基于著名的降基方法（RBM）的首次此类努力。我们的方法建立在高保真解算器的基础上，该解算器在角空间中采用离散坐标法，在物理空间中采用渐近保持迎风不连续伽辽金法，并对所得到的线性系统进行有效的综合加速源迭代。为了解决通过积分算子耦合参数值（或角方向）的挑战，我们方法的第一个新颖成分是一个迭代过程，其中宏观密度是从RBM快照构建的，明确处理并允许传输扫描，然后更新。然后利用贪心算法自适应地选择角空间中的代表性样本，形成代理解空间。第二个新颖之处是最小二乘密度重建策略，在每个相关的物理位置，实现了对任意非结构化角度样本集的宏观密度的鲁棒和精确集成。数值实验表明，该方法可以有效地降低各种情况下的计算成本。
Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The fundamental difficulty hampering their efficient and accurate numerical resolution lies in the high dimensionality of the physical and velocity/angular variables and the fact that the problem is multiscale in nature. Leveraging the existence of a hidden low-rank structure hinted by the diffusive limit, in this work, we design and test the angular-space reduced order model for the linear radiative transfer equation, the first such effort based on the celebrated reduced basis method (RBM). Our method is built upon a high-fidelity solver employing the discrete ordinates method in the angular space, an asymptotic preserving upwind discontinuous Galerkin method for the physical space, and an efficient synthetic accelerated source iteration for the resulting linear system. Addressing the challenge of the parameter values (or angular directions) being coupled through an integration operator, the first novel ingredient of our method is an iterative procedure where the macroscopic density is constructed from the RBM snapshots, treated explicitly and allowing a transport sweep, and then updated afterwards. A greedy algorithm can then proceed to adaptively select the representative samples in the angular space and form a surrogate solution space. The second novelty is a least squares density reconstruction strategy, at each of the relevant physical locations, enabling the robust and accurate integration over an arbitrarily unstructured set of angular samples toward the macroscopic density. Numerical experiments indicate that our method is effective for computational cost reduction in a variety of regimes.