A space-time-parameters structure of the parametric parabolic PDEs motivates the application of tensor methods to define reduced order models (ROMs). Within a tensor-based ROM framework, the matrix SVD -- a traditional dimension reduction technique -- yields to a low-rank tensor decomposition (LRTD). Such tensor extension of the Galerkin proper orthogonal decomposition ROMs (POD-ROMs) benefits both the practical efficiency of the ROM and its amenability for the rigorous error analysis when applied to parametric PDEs. The paper addresses the error analysis of the Galerkin LRTD-ROM for an abstract linear parabolic problem that depends on multiple physical parameters. An error estimate for the LRTD-ROM solution is proved, which is uniform with respect to problem parameters and extends to parameter values not in a sampling/training set. The estimate is given in terms of discretization and sampling mesh properties, and LRTD accuracy. The estimate depends on the smoothness rather than on the Kolmogorov n-widths of the parameterized manifold of solutions. Theoretical results are illustrated with several numerical experiments.

翻译：参数抛物型PDE的时空-参数结构促使张量方法应用于定义降阶模型(ROMs)。在基于张量的ROM框架中，矩阵SVD——传统降维技术——转化为低秩张量分解(LRTD)。这种Galerkin本征正交分解ROMs(POD-ROMs)的张量扩展既提升了ROM的实际效率，也增强了其对参数化PDE进行严格误差分析的适用性。本文针对依赖于多个物理参数的抽象线性抛物问题，分析了Galerkin LRTD-ROM的误差。证明了LRTD-ROM解的误差估计，该估计关于问题参数一致，并可推广至非采样/训练集内的参数值。估计以离散化和采样网格属性以及LRTD精度表示，其依赖的是参数化解流形的光滑性而非Kolmogorov n-宽度。通过若干数值实验验证了理论结果。