  This work proposes novel techniques for the efficient numerical simulation of
parameterized, unsteady partial differential equations. Projection-based
reduced order models (ROMs) such as the reduced basis method employ a
(Petrov-)Galerkin projection onto a linear low-dimensional subspace. In
unsteady applications, space-time reduced basis (ST-RB) methods have been
developed to achieve a dimension reduction both in space and time, eliminating
the computational burden of time marching schemes. However, nonaffine
parameterizations dilute any computational speedup achievable by traditional
ROMs. Computational efficiency can be recovered by linearizing the nonaffine
operators via hyper-reduction, such as the empirical interpolation method in
matrix form. In this work, we implement new hyper-reduction techniques
explicitly tailored to deal with unsteady problems and embed them in a ST-RB
framework. For each of the proposed methods, we develop a posteriori error
bounds. We run numerical tests to compare the performance of the proposed ROMs
against high-fidelity simulations, in which we combine the finite element
method for space discretization on 3D geometries and the Backward Euler time
integrator. In particular, we consider a heat equation and an unsteady Stokes
equation. The numerical experiments demonstrate the accuracy and computational
efficiency our methods retain with respect to the high-fidelity simulations.
