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.

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