This paper concerns thermoacoustic tomography and photoacoustic tomography, two couple-physics imaging modalities that attempt to combine the high resolution of ultrasound and the high contrast capabilities of electromagnetic waves. We give sufficient conditions to recover both the sound speed of the medium being probed and the source.

We consider reshaping an obstacle virtually by using transformation optics in acoustic and electromagnetic scattering. Among the general virtual reshaping results, the virtual minification and virtual magnification in particular are studied. Stability estimates are derived for scattering amplitude in terms of the diameter of a small obstacle, which implies that the limiting case for minification corresponds to a perfect cloaking, i.e., the obstacle is invisible to detection.

We study resonance for the Helmholtz equation with a finite frequency in a plasmonic material of negative dielectric constant in two and three dimensions. We show that the quasi-static approximation is valid for diametrically small inclusions. In fact, we quantitatively prove that if the diameter of an inclusion is small compared to the loss parameter, then resonance occurs exactly at eigenvalues of the Neumann--Poincaré operator associated with the inclusion.

By adding a reference ball as an extra artificial obstacle component to the underlying scattering system, we propose a simple but robust and effective technique to choose the crucial cut-off value required in the linear sampling method (LSM) for inverse scattering problems. The reference ball technique causes few extra computational costs to the LSM, but brings in a practically important byproduct, i.e., it eliminates the interior eigenvalue problem automatically, a well-known barrier when applying the LSM. Some mathematical justifications of the new technique are provided, and numerical experiments are also presented to demonstrate its feasibility and effectiveness.