Applied Mathematics Colloquium: Aseel Titi (JHU)

Abstract: In this talk, we will address two inverse problems where the available data is limited. The first part will focus on inverse gravimetry, where the objective is to determine the mass density distribution of a body based on gravitational measurements. Our approach deviates from traditional gravimetry, where data is typically collected over a curve or surface. Instead, we explore scenarios where data is only available at a few discrete points. Rather than aiming to reconstruct a full density distribution, our work focuses on recovering the parameters of simple geometric shapes, such as ellipses and rectangles in two dimensions, and ellipsoids and rectangular parallelepipeds in three dimensions. In the second part, we focus on Electrical Impedance Tomography (EIT), which entails solving the inverse problem of determining the conductivity distribution within a domain Ω from boundary measurements of electric currents and voltages. Specifically, our work addresses the identification of a small elliptical conductivity anomaly within a unit disc, using boundary measurements. The conductivity of the anomaly is assumed to be a small perturbation from the constant background. A key innovation in our approach is the use of electrode pairs for measurements, with the ability to move these pairs to take additional measurements. We show that three measurements are sufficient to determine the size and location of the anomaly within the disc, establishing the uniqueness and stability of the inverse problem. Furthermore, two additional measurements are necessary to determine the ellipse’s aspect ratio and orientation. The investigation also includes an analysis of the stability of the inverse problem and optimal experiment design.