Fast matrix exponential-based quasi-boundary value methods for inverse space-dependent source problems

In this paper, we study the well-established quasi-boundary value methods for regularizing inverse state-dependent source problems, where the convergence analysis of three typical cases is presented in the framework of filtering regularization method under suitable source conditions. Interestingly, the quasi-boundary value methods can be interpreted as certain Lavrentiev-type regularization, which was not known in literature. As another major contribution, efficient numerical implementation based on matrix exponential in time is developed, which shows much improved computational efficiency than MATLAB's backslash solver based on the all-at-once space-time discretization scheme. Numerical examples are reported to illustrate the promising computational performance of our proposed algorithms based on matrix exponential techniques.