Two-stage numerical method for sparse source identification problem of linear diffusion-advection equations

<p style="text-align:center;"> <strong>(Communicated by Changjun Yu)</strong> <p style="text-align:justify;"> <span> This paper addresses a sparse initial source identification problem with linear diffusion-advection equations and a given final time. The problem's complexity arises from the initial source's structure, a linear combination of Dirac measures. Moreover, the strong diffusion and smoothing effects of the linear diffusion-advection equation further render the problem exponentially ill-posed and non-smooth. To overcome these obstacles, we propose a two-stage numerical method. In the first stage, we reformulate the identification problem as a sparse optimal control problem, incorporating $L^1$ and $L^2$ regularization terms into the cost function to encourage sparsity and ensure well-posedness. To solve the sparse optimal control problem efficiently, augmented Lagrangian technique is extent to an inexact setting, where the sub-problem in each iteration is addressed using the semi-smooth Newton method. For the large-scale and ill-conditioned Newton system involved, an efficient preconditioned conjugate gradient method is designed. Additionally, we demonstrate that with the appropriate selection of the penalty parameter, the proposed inexact augmented Lagrangian algorithm can achieve super-linear convergence rate. At the second stage, we refine the solution obtained at the first stage into a linear combination of Dirac measures. This process entails a location identification procedure and solving a small-scale least squares fitting problem. Lastly, three types of sparse initial source identification problems are considered and solved to demonstrate the feasibility and effectiveness of the proposed algorithm.</span>