Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds
Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds
复制标题
张量流形上非线性演化方程的自适应积分
DOI：
10.1007/s10915-022-01868-x
复制
发表时间：
2020
影响因子：
2.5
通讯作者：
D. Venturi
中科院分区：
文献类型：
--
作者：
Abram Rodgers;Alec Dektor;D. Venturi
提出了张量流形上非线性发展方程时间积分的新的自适应算法。这些算法，我们称为步长截断方法，是基于用传统的时间步进方案执行一个时间步长，然后在张量流形上进行截断操作。通过自适应地选择张量流形的秩来满足稳定性和精度要求，我们证明了各种步长截断方法的收敛，包括显式一步和多步方法。这些方法非常容易实现，因为它们只依赖张量之间的算术运算，而这些运算可以通过高效且可扩展的并行算法来执行。自适应步长截断方法可用于计算高维偏微分方程组的数值解，已成为许多新的应用领域的核心，如最优质量传输、随机动力系统和平均场最优控制。给出并讨论了二维和四维平面环面上具有空间相关漂移的Fokker-Planck方程的数值应用。
We develop new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional time-stepping scheme, followed by a truncation operation onto a tensor manifold. By selecting the rank of the tensor manifold adaptively to satisfy stability and accuracy requirements, we prove convergence of a wide range of step-truncation methods, including explicit one-step and multi-step methods. These methods are very easy to implement as they rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Adaptive step-truncation methods can be used to compute numerical solutions of high-dimensional PDEs, which, have become central to many new areas of application such optimal mass transport, random dynamical systems, and mean field optimal control. Numerical applications are presented and discussed for a Fokker-Planck equation with spatially dependent drift on a flat torus of dimension two and four.