Existence of dynamical low-rank approximations to parabolic problems

一种用于高维抛物型偏微分方程的时空自适应低秩方法

A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations

本研究提出一种结合时间稀疏小波展开与空间自适应低秩近似的高维抛物型偏微分方程自适应方法，证明其收敛性与复杂度优势，还可生成可计算的严格后验误差界，并通过数值实验验证结果。

高维抛物问题的动态低秩张量逼近：空间离散化的存在性与收敛性

Dynamical low-rank tensor approximations to high-dimensional parabolic problems: existence and convergence of spatial discretizations

本文针对高维抛物问题提出动态低秩张量逼近框架，证明解的存在性、数据扰动稳定性及空间离散化收敛性，并兼容 tensor train 与分层张量等格式。

关于低秩投影-分裂积分器在双曲和抛物方程中的稳定性研究

On the stability of the low-rank projector-splitting integrators for hyperbolic and parabolic equations

本文研究了低秩投影-分裂积分器（PSI）在双曲和抛物方程中的稳定性，通过von Neumann分析，探讨了低秩时间积分器与标准空间离散化方法结合的稳定性，揭示了分裂方法对稳定性区域的影响。

Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations

We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a parabolic type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in cer…

Efficient low rank approximations for parabolic control problems with unknown heat source

An inverse problem of finding an unknown heat source for a class of linear parabolic equations is considered. Such problems can typically be converted to a direct problem with non-local conditions in time instead of an initial value problem. Standard ways of solving these non-local problems include direct temporal and spatial discretization as well as the shooting method, which may be computationally expensive in higher dimensions. In the present article, we present approaches based on low-rank approximation via Arnoldi algorithm to bypass the computational limitations of the mentioned classic…

三维抛物型方程的低秩张量列车有限差分方法

The low-rank tensor-train finite difference method for three-dimensional parabolic equations

本文提出一种基于张量列车重构二阶有限差分法的三维抛物型问题低秩近似数值框架，结合显隐式Euler和Crank-Nicolson方法推进时间，并采用无矩阵PCG求解线性系统，在保证精度的同时大幅节省时空开销，适用于高维问题。