The paper addresses the problem of finding a low-rank approximation of a multi-dimensional tensor, $\Phi $, using a subset of its entries. A distinctive aspect of the tensor completion problem explored here is that entries of the $d$-dimensional tensor $\Phi$ are reconstructed via $C$-dimensional slices, where $C < d - 1$. This setup is motivated by, and applied to, the reduced-order modeling of parametric dynamical systems. In such applications, parametric solutions are often reconstructed from space-time slices through sparse sampling over the parameter domain. To address this non-standard completion problem, we introduce a novel low-rank tensor format called the hybrid tensor train. Completion in this format is then incorporated into a Galerkin reduced order model (ROM), specifically an interpolatory tensor-based ROM. We demonstrate the performance of both the completion method and the ROM on several examples of dynamical systems derived from finite element discretizations of parabolic partial differential equations with parameter-dependent coefficients or boundary conditions.

翻译：本文研究如何利用多维张量$\Phi$的部分元素来寻找其低秩近似。本研究所探讨的张量补全问题的一个显著特点是：$d$维张量$\Phi$的元素通过$C$维切片进行重构，其中$C < d - 1$。该框架的建立受到参数化动力系统降阶建模的启发并应用于该领域。在此类应用中，参数化解通常通过参数域上的稀疏采样从时空切片中重构。针对这一非标准补全问题，我们提出了一种称为混合张量链的新型低秩张量格式。随后将该格式下的补全方法融入Galerkin降阶模型（ROM），具体而言是一种基于插值的张量ROM。我们通过多个动力系统实例验证了补全方法与ROM的性能，这些实例源自具有参数依赖系数或边界条件的抛物型偏微分方程的有限元离散化。