Fast numerical approximation of parabolic problems using model order reduction and the Laplace transform

We introduce a method for the fast numerical approximation of linear, second-order parabolic partial differential equations (PDEs for short) with time-independent coefficients based on model order reduction techniques and the Laplace transform. We start by applying this transform to the evolution problem, thus yielding a time-independent boundary value problem solely depending on the complex Laplace variable. In an offline stage, we judiciously sample the Laplace variable 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 consists of using 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 illustrate 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.