  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.
