AI 中文总结
研究刚性、与时间相关的常微分方程组，提出在张量列车格式内求解的通用框架，开发KIOPS-TT和RK2EXPINT-TT，通过重新制定积分器方案及增强刚度张量，利用克雷洛夫子空间方法实现加速，数值实验验证了低秩场景下的显著提速。
AI 中文摘要
微分方程在众多应用中出现，尤其是在科学和技术领域。本文聚焦于刚性、与时间相关的常微分方程组。指数积分器旨在通过精确积分线性部分并通过φ函数的线性组合近似非线性部分来求解此类方程。现有方法利用增广刚度矩阵一步求解线性部分并评估非线性部分的φ函数线性组合。但经典方法假设系统由矩阵和向量表示，未利用潜在的高维结构。张量可解决此限制。本文提供了在张量列车（TT）格式内直接求解刚性、与时间相关系统的通用框架，开发了KIOPS-TT和RK2EXPINT-TT，重新制定了显式指数龙格-库塔积分器的张量方案，增强刚度张量以通过克雷洛夫子空间方法单次评估指数函数来计算作用于张量的φ函数线性组合。还表明矩阵方法的基础理论在张量情况下仍然有效。数值实验证实，在低秩场景中，KIOPS-TT和RK2EXPINT-TT比经典对应方法有显著加速。
英文摘要
Differential equations arise in numerous applications, particularly within scientific and technical contexts. Systems of stiff, time-dependent ordinary differential equations constitute the focus of this work. Exponential integrators are designed to solve such equations by integrating the linear part exactly, while simultaneously approximating the nonlinear part through a linear combination of $φ$-functions. By utilizing an augmented stiffness matrix, state-of-the-art methods like KIOPS and RK2EXPINT solve the linear part and evaluate linear combinations of $φ$-functions for the nonlinear part in a single step, effectively reducing the computational effort to a single matrix exponential evaluation. However, these classical approaches assume that the system is represented by matrices and vectors, potentially not utilizing the underlying high-dimensional structure. Tensors address this limitation and offer significant storage efficiency through well-established decompositions like the Tensor Train (TT) format. This work provides a general framework for solving stiff, time-dependent systems directly within the TT format. Specifically, KIOPS-TT and RK2EXPINT-TT are developed as extensions of the original KIOPS and RK2EXPINT algorithms. This involves reformulating the scheme of explicit exponential Runge-Kutta integrators for tensors and augmenting the stiffness tensor to compute linear combinations of $φ$-functions acting on tensors through a single evaluation of the exponential function using Krylov subspace methods. Furthermore, it is shown that the underlying theory of the matrix methods remains valid, thereby enabling the transfer of key theorems to the tensor case. Numerical experiments confirm significant speed-ups for KIOPS-TT and RK2EXPINT-TT in low-rank scenarios compared to their classical counterparts.
