Hybrid Inverse Problems and Internal Functionals
Guillaume Bal, Columbia University

Several recent coupled-physics medical imaging modalities aim to combine a high-contrast, low-resolution, modality with a high-resolution, low-contrast, modality and ideally offer high-contrast, high-resolution, reconstructions. Mathematically, these modalities involve the reconstruction of constitutive parameters in partial differential equations from knowledge of internal functionals of the parameters and solutions to said equations. This recent field of research is often referred to as Hybrid Inverse Problems.

This talk presents recent theoretical results of uniqueness, stability and explicit reconstructions for several hybrid inverse problems. We provide an explicit characterization of what can (and cannot) be reconstructed in coupled-physics imaging modalities such as Magnetic Resonance Elastography, Transient Elastography, Photo-Acoustic Tomography, and Ultrasound Modulation Tomography. Numerical simulations confirm the high-resolution, high-contrast, potential of these novel modalities.