We consider dynamical low-rank approximations to parabolic problems on higher-order tensor manifolds in Hilbert spaces. In addition to existence of solutions and their stability with respect to perturbations to the problem data, we show convergence of spatial discretizations. Our framework accommodates various standard low-rank tensor formats for multivariate functions, including tensor train and hierarchical tensors.

翻译：我们考虑希尔伯特空间中高阶张量流形上抛物问题的动态低秩逼近。除了解的存在性及其对问题数据扰动的稳定性外，我们还证明了空间离散化的收敛性。我们的框架适用于多变量函数的各种标准低秩张量格式，包括张量列和层次张量。