2018 Hellman Fellow
Assistant Professor Mathematics
UC Santa Cruz
Project Title: Inverse problems, integral geometry and uncertainty quantification
Project Description: Several problems of medical and geophysical imaging are modeled as ‘inverse problems in integral geometry’, where the unknown to be reconstructed is an internal parameter and the measurements consist of integral functionals of the unknown. A prototypical example of this occurs in every day’s CAT scan, which reconstructs a patient’s tissue density from its integrals along a family of straight lines through the patient’s body.
For the specific model just mentioned, we know very well how to answer the following questions: is reconstruction possible ? If yes, how to do it ? If the measurements are noisy (they certainly are !), how to guarantee that the reconstruction approach is statistically optimal ?
Professor Monard’s research is concerned with the mathematical and numerical analysis of several generalizations of the problem above, for which the questions just formulated remain actively studied at the moment. Such generalizations involve integration along more general families of curves (modeling for example the trajectories of photons in a medium with variable refractive index), more general integrands (e.g., vector/tensor fields), and non-linear integral geometric problems such as the travel-time tomography problem.