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NJIT Applied Mathematics Colloquium

Friday, February 11 2011, 11:30am
Cullimore Lecture Hall II
New Jersey Institute of Technology

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Inverse elliptic problems with internal controls and applications to Hybrid Imaging

Guillaume Bal

Columbia

Electrical Impedance Tomography (EIT) and Optical Tomography (OT) are useful medical imaging modalities because of the high contrasts displayed between the electrical and optical responses 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 allow us to combine large contrasts and high resolution. Mathematically, these modalities require that one solve inverse problems of partial differential equations (PDE) from knowledge of internal functionals of the PDE solutions. 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.