In this talk, we briefly talk about the inverse boundary/medium problems with the semilinear transport model which naturally rises from two-photon absorption optics. The model can be formally derived from a paraxial setting of a nonlinear absorption wave model. For the related inverse problems, we consider two cases. For the inverse boundary problem, we adopted the usual linearization technique and prove the uniqueness of reconstruction. For the inverse medium problems, we consider the problem from photoacoustic imaging specifically in static and time-dependent settings and prove the uniqueness of the reconstruction for the absorption coefficients.

In this talk, we will discuss an inverse problem of determining the spatially dependent source term and the Robin boundary coefficient in a time-fractional diffusion equation, with the aid of extra measurement data at a subdomain near the accessible boundary. Firstly, the spatially varying source is uniquely determined in view of the unique continuation principle and Duhamel principle for the fractional diffusion equation. The Hopf lemma for a homogeneous time-fractional diffusion equation is proved and then used to prove the uniqueness of recovering the Robin boundary coefficient. Numerically, based on the theoretical uniqueness result, we apply the classical Tikhonov regularization method to transform the inverse problem into a minimization problem, which is solved by an iterative thresholding algorithm. Finally, several numerical examples are presented to show the accuracy and effectiveness of the proposed algorithm.