  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.
