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.

The recently developed data-driven eigenmatrix method shows very promising reconstruction accuracy in sparse recovery for a wide range of kernel functions and random sample locations. However, its current implementation can lead to numerical instability if the threshold tolerance is not appropriately chosen. To incorporate regularization techniques, we have proposed to regularize the eigenmatrix method by replacing the computation of an ill-conditioned pseudo-inverse by the solution of an ill-conditioned least squares system, which can be efficiently treated by Tikhonov regularization. Extensive numerical examples confirmed the improved effectiveness of our proposed method, especially when the noise levels were relatively high.