论文标题
在张量流形上非线性演化方程的自适应整合
Adaptive integration of nonlinear evolution equations on tensor manifolds
论文作者
论文摘要
我们开发了新的自适应算法，用于在张量歧管上的非线性演化方程的时间整合。这些算法（我们称之为步进方法方法）基于使用常规的时间步骤方案执行一个时间步骤，然后在张量歧管上进行截断操作。通过自适应地选择张量歧管的等级以满足稳定性和准确性要求，我们证明了广泛的阶梯截断方法的收敛性，包括明确的一步和多步方法。这些方法非常容易实现，因为它们仅依赖于张量之间的算术操作，这些操作可以通过有效且可扩展的并行算法执行。自适应级别截断方法可用于计算高维PDE的数值解，这些解决方案已成为许多新应用领域的核心，例如最佳的质量传输，随机动力学系统和平均野外最佳控制。为Fokker-Planck方程式提供了数值应用，并讨论了在尺寸二和四的平坦圆环上的空间依赖漂移。
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.
