  We introduce a novel, fast method for the numerical approximation of
parabolic partial differential equations (PDEs for short) based on model order
reduction techniques and the Laplace transform. We start by applying said
transform to the evolution problem, thus yielding a time-independent boundary
value problem solely depending on the complex Laplace parameter. In an offline
stage, we judiciously sample the Laplace parameter and numerically solve the
corresponding collection of high-fidelity or full-order problems. Next, we
apply a proper orthogonal decomposition (POD) to this collection of solutions
in order to obtain a reduced basis in the Laplace domain. We project the linear
parabolic problem onto this basis, and then using any suitable time-stepping
method, we solve the evolution problem. A key insight to justify the
implementation and analysis of the proposed method corresponds to resorting to
Hardy spaces of analytic functions and establishing, through the Paley-Wiener
theorem, an isometry between the solution of the time-dependent problem and its
Laplace transform. As a result, one may conclude that computing a POD with
samples taken in the Laplace domain produces an exponentially accurate reduced
basis for the time-dependent problem. Numerical experiments portray the
performance of the method in terms of accuracy and, in particular, speed-up
when compared to the solution obtained by solving the full-order model.
