Test calendar

Inside Out: Seeing Inside the Unknown with Boundary Measurements 

Abstract: How much can we learn about the interior of an object of interest, such as the human body or planet Earth, without breaking it open? An inverse boundary problem seeks to determine the internal properties of a medium, represented by the coefficients of a PDE, from measurements taken at its boundary. Such problems arise in a wide range of applications, including medical imaging, non-destructive industrial testing, and seismic imaging.

In this talk, we will discuss recent advancements in inverse problems for elliptic, time-dependent Schrödinger, and hyperbolic equations. In particular, we show that boundary measurements can be used to recover magnetic and electric potentials for the magnetic Schrödinger operator, identify parameters governing atomic interactions in the nonlinear Gross–Pitaevskii equation, and determine magnetic and electric potentials in a Riemannian wave operator Hölder stably. The central theme of this talk is the development of robust mathematical tools to address key challenges, including the recovery of information from partial boundary data, the handling of nonlinear couplings, and the derivation of stability estimates.