UC Riverside

In a joint effort with my advisor, we study stability of reconstruction in current density impedance imaging (CDII), that is, the inverse problem of recovering the conductivity of a body from the measurement of the magnitude of the current density vector field in the interior of the object. Our results show that CDII is stable with respect to errors in interior measurements of the current density vector field, and confirm the stability of reconstruction which was previously observed in numerical simulations, and was long believed to be the case. Next, we show that CDII is stable with respect to errors in both measurement of the magnitude of the current density vector field in the interior and the measurement of the voltage potential on the boundary. This completes the authors study of the global stability of Current Density Independence Imaging. These results are accomplished through analysis on a related functional from the so-called least gradient problem as well as geometric arguments on the level sets of the induced voltage potential function. These geometric arguments are dependent upon some ad hoc conditions which are shown to be guaranteed by reasonable sufficient conditions.

Additionally, we study the Inverse Sturm-Liouville problem which is the problem of reconstructing the coefficient function $q$ from the second order elliptic differential operator $-\nabla + q$ using the boundary spectral data. While there are several results in one dimension and higher dimensions using complete spectral data and even finitely many terms omitted, none have explored results for a subsequence of spectral data. We aim to establish such results in one dimension and higher dimensions by using the asymptotic behavior of eigenfunctions on the boundary.