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.

翻译：我们提出了一种基于模型降阶技术和拉普拉斯变换的新颖快速数值逼近方法，用于求解抛物型偏微分方程。首先对演化问题应用拉普拉斯变换，从而得到仅依赖于复拉普拉斯参数的时间无关边值问题。在离线阶段，我们合理采样拉普拉斯参数，并数值求解相应的高保真或全阶问题集合。接着，对该解集合应用本征正交分解（POD），以获得拉普拉斯域中的降阶基。将线性抛物型问题投影到该基上后，采用任意合适的时间步进方法求解演化问题。为证明所提方法的可行性并支撑其分析，关键思路在于借助解析函数的Hardy空间，并通过Paley-Wiener定理建立时变问题解与其拉普拉斯变换之间的等距关系。由此可得出结论：利用拉普拉斯域中的采样进行POD计算，可为时变问题生成指数精度的降阶基。数值实验展示了该方法在精度方面的性能，特别是与全阶模型求解相比的加速效果。