A space-time-parameters structure of 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 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 local smoothness rather than on the Kolmogorov n-widths of the parameterized manifold of solutions. Theoretical results are illustrated with several numerical experiments.

翻译：参数化抛物型偏微分方程所具有的时空参数结构，促使我们应用张量方法来定义降阶模型。在基于张量的降阶模型框架中，作为传统降维技术的矩阵奇异值分解被推广为低秩张量分解。这种对Galerkin本征正交分解降阶模型的张量扩展，不仅提升了降阶模型的实际计算效率，而且使其在应用于参数化偏微分方程时更易于进行严格的误差分析。本文针对一个依赖多个物理参数的抽象线性抛物型问题，分析了Galerkin低秩张量分解降阶模型的误差。我们证明了低秩张量分解降阶模型解的一个误差估计，该估计关于问题参数具有一致性，并可推广至未出现在采样/训练集中的参数值。该估计由离散化网格特性、采样网格特性以及低秩张量分解精度共同决定。值得注意的是，该估计依赖于解流形在参数化过程中的局部光滑性，而非解流形的Kolmogorov n-宽度。理论结果通过若干数值实验得到了验证。