Yang Zhang (UW)

Friday, January 19, 2024 - 9:30pm to 10:30pm

PDL C-401

Microlocal analysis provides a framework to study distributions (generalized functions) and operators from a phase space point of view. It is rooted in Fourier analysis and further developed by functional analysis and symplectic geometry. A fundamental concept in this field, singularities, refer to points in phase space where a distribution is not smooth.

In this talk, we will discuss the study of several inverse problems using microlocal analysis, especially from the perspective of singularities. The first inverse problem is to reconstruct a density function of an object from its integral transform over cone surfaces, which arises in Compton camera imaging. Using microlocal analysis, we describe which features (singularities) of the object can be reconstructed in a stable way from local data measurements. Next, we will talk about inverse problems of recovering parameters in partial differential equations from boundary measurements, especially addressing nonlinear wave equations in ultrasound imaging with damping effects. We will explain how the analysis of new singularities, produced by the nonlinear interactions of waves, enables the determination of these parameters. In particular, we will address the challenges in inverse boundary value problems and the techniques to handle them.