IP@AM's primary focus is the analysis of Inverse Problems related to Partial Differential Equations (PDE), typically the reconstruction of the constitutive parameters in these equations from available measurements, such as measurements of the PDE solution at the boundary of a domain of interest, or when they are available inside the domain of interest. Such PDEs include elliptic equations, transport equations, Wave equations, Newton's equations. Applications for such theories are found primarily in Medical Imaging, Geophysical Imaging and Biological Imaging.
Theoretical and Numerical questions of interest in the group include:
- Uniqueness of the reconstructions; What it is we can or cannot reconstruct in the absence of noise
- Stability of the reconstructions; How small modifications in the range of the measurement operator affect the reconstruction
- Modeling of noise in the PDE model and/or in the measurement operator; How are reconstructions affected by physically motivated noise
- Numerical tools tailored for solutions of inverse problems; Well-posed inverse problems typically rely on singular solutions of the PDE. How can one simulate such singular solutions numerically.
For a list of preprints written by members in the group, see here.Examples of applications include:
- reconstruction of the optical parameters (absorption and scattering) in transport theory. Applications in medical (CT, optical tomography...) and atmospheric imaging.
- reconstruction of the metric of a manifold from travel-time measurements along a simple class of curves. Applications in geophysical imaging.
- reconstruction of wave-emitting inclusions in a random medium from boundary measurements. Applications in detection.
- reconstruction of the electric potential and the magnetic field from scattering data in the context of the relativistic Newton equation.