  The paper introduces a reduced order model (ROM) for numerical integration of
a dynamical system which depends on multiple parameters. The ROM is a
projection of the dynamical system on a low dimensional space that is both
problem-dependent and parameter-specific. The ROM exploits compressed tensor
formats to find a low rank representation for a sample of high-fidelity
snapshots of the system state. This tensorial representation provides ROM with
an orthogonal basis in a universal space of all snapshots and encodes
information about the state variation in parameter domain. During the online
phase and for any incoming parameter, this information is used to find a
reduced basis that spans a parameter-specific subspace in the universal space.
The computational cost of the online phase then depends only on tensor
compression ranks, but not on space or time resolution of high-fidelity
computations. Moreover, certain compressed tensor formats enable to avoid the
adverse effect of parameter space dimension on the online costs (known as the
curse of dimension). The analysis of the approach includes an estimate for the
representation power of the acquired ROM basis. We illustrate the performance
and prediction properties of the ROM with several numerical experiments, where
tensorial ROM's complexity and accuracy is compared to those of conventional
POD-ROM.
