Time-dependent kinetic models are ubiquitous in computational science and engineering. The underlying integro-differential equations in these models are high-dimensional, comprised of a six--dimensional phase space, making simulations of such phenomena extremely expensive. In this article we demonstrate that in many situations, the solution to kinetics problems lives on a low dimensional manifold that can be described by a low-rank matrix or tensor approximation. We then review the recent development of so-called low-rank methods that evolve the solution on this manifold. The two classes of methods we review are the dynamical low-rank (DLR) method, which derives differential equations for the low-rank factors, and a Step-and-Truncate (SAT) approach, which projects the solution onto the low-rank representation after each time step. Thorough discussions of time integrators, tensor decompositions, and method properties such as structure preservation and computational efficiency are included. We further show examples of low-rank methods as applied to particle transport and plasma dynamics.

翻译：时间依赖动力学模型在计算科学与工程领域普遍存在。这些模型中的基础积分-微分方程具有高维特性，由六维相空间构成，使得对此类现象的模拟计算成本极高。本文论证了在许多情况下，动力学问题的解存在于低维流形上，该流形可通过低秩矩阵或张量近似进行描述。随后，我们综述了近年来发展的低秩方法，这些方法可在该流形上演化解。我们重点评述的两类方法包括：动态低秩方法（通过推导低秩因子的微分方程实现）和步进截断方法（在每个时间步后将解投影至低秩表示）。本文系统探讨了时间积分器、张量分解以及结构保持性、计算效率等方法特性，并展示了低秩方法在粒子输运和等离子体动力学中的具体应用案例。