A fast reduced order method for linear parabolic inverse source problems
Provides an efficient numerical solution for inverse source problems in parabolic PDEs, which are important in engineering and physics.
Proposed a reduced order method for linear parabolic inverse source problems that achieves computational savings compared to finite element method while maintaining accuracy, with rigorous error estimates.
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.