In this paper, we propose a novel, computationally efficient reduced order method to solve linear parabolic inverse source problems. Our approach provides accurate numerical solutions without relying on specific training data. The forward solution is constructed using a Krylov sequence, while the source term is recovered via the conjugate gradient (CG) method. Under a weak regularity assumption on the solution of the parabolic partial differential equations (PDEs), we establish convergence of the forward solution and provide a rigorous error estimate for our method. Numerical results demonstrate that our approach offers substantial computational savings compared to the traditional finite element method (FEM) and retains equivalent accuracy.

翻译：本文提出一种新颖且计算高效的降阶方法，用于求解线性抛物型逆源问题。该方法无需依赖特定训练数据即可提供精确数值解。正向解通过Krylov序列构造，而源项则采用共轭梯度（CG）方法恢复。在抛物型偏微分方程（PDEs）解满足弱正则性假设的条件下，我们建立了正向解的收敛性，并给出了本方法的严格误差估计。数值结果表明，与传统有限元法（FEM）相比，本方法在保持同等精度的同时，显著降低了计算成本。