Identifying source term in the subdiffusion equation with L^2-TV regularization

TL;DR

This paper addresses the inverse source problem in subdiffusion equations by introducing an L^2-TV regularization approach, providing theoretical error estimates, and developing an efficient primal-dual algorithm validated through numerical tests.

Contribution

It proposes a novel L^2-TV regularization method for the inverse source problem in subdiffusion equations, with theoretical analysis and an accelerated algorithm.

Findings

The regularized solution converges to the true source as noise decreases.

The proposed algorithm is efficient and accurate in numerical experiments.

Error estimates demonstrate the method's reliability.

Abstract

In this paper, we consider the inverse source problem for the time-fractional diffusion equation, which has been known to be an ill-posed problem. To deal with the ill-posedness of the problem, we propose to transform the problem into a regularized problem with L^2 and total variational (TV) regularization terms. Differing from the classical Tikhonov regularization with L^2 penalty terms, the TV regularization is beneficial for reconstructing discontinuous or piecewise constant solutions. The regularized problem is then approximated by a fully discrete scheme. Our theoretical results include: estimate of the error order between the discrete problem and the continuous direct problem; the convergence rate of the discrete regularized solution to the target source term; and the convergence of the regularized solution with respect to the noise level. Then we propose an accelerated primal-dual iterative algorithm based on an equivalent saddle-point reformulation of the discrete regularized model. Finally, a series of numerical tests are carried out to demonstrate the efficiency and accuracy of the algorithm.