论文标题
高维非线性PDE的等级自适应张量方法
Rank-adaptive tensor methods for high-dimensional nonlinear PDEs
论文作者
论文摘要
我们提出了一种新的等级自适应张量方法，以计算高维非线性PDE的数值解。该方法结合了功能张量列车（FTT）系列膨胀，操作员分裂时间积分以及基于阈值标准的新级别自适应算法，该算法限制了PDE Velicity Velicity Velocity velicity velicity formor的组件到FTT张量歧管。这产生了一个方案，可以随着时间积分进行时从PDE解决方案中添加或删除张量模式。该新方法旨在提高高维问题的数值整合中的计算效率，准确性和鲁棒性。特别是，它克服了与动态张量集成相关的众所周知的计算挑战，包括低级别的建模误差以及每个时间步骤中张量内核的协方差矩阵的需求。在两个维度和四维Fokker-Planck方程中，对线性和非线性对流问题进行了数字应用和讨论。
We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical integration of high-dimensional problems. In particular, it overcomes well-known computational challenges associated with dynamic tensor integration, including low-rank modeling errors and the need to invert covariance matrices of tensor cores at each time step. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker-Planck equation.
