UC Riverside

This dissertation investigates two inverse problems, one on electrical networks and another from photo acoustic tomography. First we consider the inverse problem of recovering the conductivities of an electrical network from the

knowledge of the magnitude of the current along the edges coupled with either the voltage on the boundary of the

network or the current flowing in or out of the network. This problem corresponds to finding the minimizers of a l^1

minimization problem. Additionally, we show that while the conductivities are not determined uniquely the flow of the

current is uniquely determined. We will also present a convergent numerical algorithm for solving these problems along

with basic numerical simulations. Lastly, we will discuss some applications of this inverse problem.

Next we consider the inverse problem of determining both the source of a wave and its speed inside a medium

from measurements of the solution of the wave equation on the boundary. This problem arises in photoacoustic and

thermoacoustic tomography. We will present a brief overview of previous uniqueness results and then present our two

original uniqueness results. If the reciprocal of the wave speed squared is harmonic in a simply connected region and

identically one elsewhere then a wave speed satisfying a natural admissibility assumption can be uniquely determined from the solution of the wave equation on the boundary of domain without knowledge of the source. If the wave speed is known and only assumed to

be bounded, then, under the same admissibility assumption, the source of the wave can be uniquely determined from

boundary measurements.