In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems efficiently. Our method consists of offline and online stages. In the off-line stage, we explore the low-dimensional structures in the solution space of the parabolic partial differential equations (PDEs) in the forward problem with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problem extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in the parabolic inverse source problems, which is referred to as the POD algorithm in this paper. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD algorithm in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD algorithm for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the POD algorithm provides considerable computational savings over the finite element method.
翻译:在本文中,我们提出一种数据驱动模型减少方法,以有效解决抛物线反源问题。我们的方法由离线和在线阶段组成。在离线阶段,我们探索在某一源函数类别前期问题的抛光部分方程式(PDEs)的解决方案空间中的低维结构,并构建少量适当的正方形分解(POD)功能,以达到显著的减少维度。用 POD 基础函数,我们可以在在线阶段非常迅速地解决前方问题。因此,我们开发一种快速算法,以解决抛光反源问题中的优化问题,本文中称之为POD算法。在对parboli PDEs解决办法的模糊常规假设下,我们证明POD算法在解决前方抛光点PDEs时的一致。此外,我们获得了POD对parbolic反源问题POD算法的错误估计值。最后,我们提供了数字示例,以证明拟议方法的准确性和效率。我们的数字结果显示,POD 定数方法的可相当程度的计算。