Dynamical low-rank approximation for radiation transport with random input

我的项目重点是开发一种有效的方法来计算不确定条件下的辐射输运。在这一应用领域，寻求依赖于高维相空间的粒子分布。这种高维相空间的数值离散化在传统方法中导致了相当大的数值代价和较高的存储要求。特别是在我们所研究的放射治疗领域，对于精度的高要求和对不确定性的考虑会产生高维的相空间，这是传统方法所不能解决的，在这个项目中，我们发展了一种基于动态低阶近似的离散化方法，大大降低了内存需求和数值代价。这种方法的效率基于三个要求，这三个要求将在项目中进行调查和开发。这些要求包括依赖于问题的演化方程的推导，稳健的空间离散化，以及现代计算机体系结构的有效实现。

The focus of my project is the development of an efficient method for the computation of radiation transport under uncertainties. In this application field, a partical distribution is sought that depends on a phase space with high dimension. The numerical discretization of this high-dimensional phase space leads to considerable numerical costs and high memory requirements in conventional methods. In particular, in the field of radiotherapy, which we study in this project, the high requirements for accuracy and the consideration of uncertainties yield phase spaces with significantly high dimension, which conventional methods cannot resolve.In this project, we develop a discretization method based on the dynamic low-rank approximation that greatly reduces memory requirements and numerical costs. The efficiency of this method is based on three requirements, which will be investigated and developed within the project. These equirements consist of a problem-dependent derivation of the evolution equations, a robust spatial discretization, and an efficient implementation for modern computer architecture.