Adaptive source term iteration : a stable formulation for radiative transfer
Adaptive source term iteration : a stable formulation for radiative transfer
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自适应源项迭代：辐射传输的稳定公式
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发表时间：
2019
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通讯作者：
M. Torrilhon
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作者：
Felix Gruber;W. Dahmen;M. Torrilhon
辐射传递问题是一个用来描述粒子在介质中运动的模型，粒子可能与介质相互作用。它被广泛应用于核物理、医学成像和天体物理等领域。从数值的角度来看，这是一个具有挑战性的问题，因为它的传输特性和相对较高的维度，具有2d−1维的解决方案（d空间和d−1方向维度）。方向域上的积分算子引入了所有方向的全局耦合，这进一步使高维复杂化。解决辐射传输问题传统上是使用非确定性蒙特卡罗方法或确定性求解器，如矩量法和离散纵坐标法。这些确定性方法通常使用相当强的假设来获得在现实物理环境中可能不成立的离散误差的先验估计。在这篇论文中，我们提出了一个新的确定性方法来求解辐射传输问题，该方法对离散解给出了严格的后验误差估计。该方法是基于一个理想的不动点迭代在一个无限维的设置，解决了动态更新的精度近似。因此，我们称这种新方法为自适应源项迭代或简称为阿斯蒂。使用后验误差估计，使我们能够解决问题，不太经常的解决方案，也降低了计算成本，通过使用自适应选择的网格。与现有的源项迭代方法的主要区别在于，阿斯蒂在迭代过程中在空间和方向域中适应空间。这样，我们可以控制迭代的误差，以保证收敛到精确解。对于输运求解器，我们使用Broersen，Dahmen和史蒂文森的不连续Petrov-Galerkin（DPG）方法。它非常适合于我们从自适应源项迭代得到的线性迁移问题，并给出可靠的后验误差估计。这是基于Banach-Nečas-Babuška稳定性理论，该理论以inf-sup估计的存在性为中心。所有这些基于后验误差估计的自适应理论在辐射传输问题中是新的。随着分析变得越来越复杂，我们还必须解决新的实现挑战。这一点在网格管理中尤其明显，网格管理涉及将不同自适应细化网格上的运输解决方案结合起来。我们的自适应源项迭代的实现是建立在通用沙丘DPG库，该库扩展了自适应散射近似的代码，并结合生活在不同的适应网格的解决方案。最后，我们给出了使用阿斯蒂实现计算的两个示例问题，说明了自适应性如何将离散公式的大小保持在可行范围内，同时保证经过验证的误差界。
The radiative transfer problem is a model used to describe particles moving in a medium with which the particles might interact. It is used in a broad variety of fields including nuclear physics, medical imaging and astrophysics. From a numerical perspective, it is a challenging problem, due to its transport character and relatively high dimensionality with a 2d−1 dimensional solution (d spatial and d−1 directional dimensions). An integral operator over the directional domain introduces a global coupling of all directions that further complicates the high dimensionality. Solving the radiative transfer problem is traditionally done either using the nondeterministic Monte Carlo method or with deterministic solvers like the method of moments and the discrete ordinates method. Those deterministic methods usually use rather strong assumptions to obtain a priori estimates on the discretization error that might not hold in realistic physical settings. In this thesis, we propose a new deterministic method for solving the radiative transfer problem that gives rigorous a posteriori error estimates on the discrete solution. This method is based on an ideal fixed-point iteration in an infinite-dimensional setting that is solved approximately with dynamically updated accuracy. Thus, we call this new method Adaptive Source Term Iteration or ASTI for short. The use of a posteriori error estimates allows us to solve problems with less regular solutions and also reduces the computational costs by using adaptively chosen grids. The main difference with regard to existing Source Term Iteration methods, which iterate in fixed discrete spaces, is that ASTI adapts the spaces, in both the spatial and directional domain, during the iteration. This way, we can control the error of our iteration to guarantee convergence towards the exact solution. For the transport solver, we use a Discontinuous Petrov–Galerkin (DPG) method from Broersen, Dahmen and Stevenson. It is well suited for the kind of linear transport problems we obtain from the Adaptive Source Term Iteration and gives reliable a posteriori error estimates. This is based on Banach–Nečas–Babuška stability theory which centers around the existence of inf-sup estimates. All this adaptivity theory based upon a posteriori error estimators is new in the context of radiative transfer problems. As the analysis gets more involved, we also have to solve new implementational challenges. This can especially be seen in the grid management which involves combining transport solutions living on different adaptively refined grids. Our implementation of the Adaptive Source Term Iteration is built upon the general-purpose Dune-DPG library which was extended by code for an adaptive scattering approximation and for combining solutions living on differently adapted grids. Finally, we give two example problems computed with our ASTI implementation which illustrate how the adaptivity keeps the size of the discretized formulation in a feasible range while guaranteeing certified error bounds.