Inverse Elliptic Problems with Internal Controls and Applications to 
Hybrid Imaging

Electrical Impedance Tomography (EIT) and Optical Tomography (OT) are 
useful medical imaging modalities because of the high contrast between 
the electrical and optical properties of healthy tissues and those of 
non-healthy tissues. However, both modalities suffer from poor 
resolution. Ultrasound tomography enjoys very high resolution 
capabilities but often suffers from low contrast. Photo-acoustic 
tomography (PAT) and ultrasound-modulated EIT or OT are novel imaging 
techniques that combine high contrast and high resolution. 
Mathematically, these modalities require that one solve inverse problems 
of partial differential equations from knowledge of internal functionals 
of the solutions to the equations. This talk will present uniqueness and 
stability results in the context of elliptic equations. Such results 
hinge on non-trivial qualitative properties of solutions to the elliptic 
PDEs often obtained by means of the so-called complex geometrical optics 
solutions. We present theoretical and numerical evidence that these 
hybrid modalities offer superior reconstructions to the "single" 
modalities taken separately.