On the inverse gravimetry problem with minimal data

Abstract

In this dissertation we considered the inverse source problem $\Delta u = \mu,$ where lim $u (x) = 0$ as |x| goes to $\infty$ and $\mu$ is zero outside a bounded domain $\Omega$. The inverse problem of gravime- try is to find $\mu$ given $\Delta \mu$ on $\partial \Omega$. Due to nonuniqueness of $\mu$ we assumed that $\mu = \chi^{(D)}$ where D is unknown domain inside $\Omega$ We first studied the two-dimensional case where we found that about five parameters of the unknown D can be stably determined given data noise in practical situations. An ellipse is uniquely determined by five parameters. We proved uniqueness and stability of recovering an ellipse for the inverse problem from minimal amount of data which are the gravitational force at three boundary points. In the proofs we derived and used simple systems of linear and non linear algebraic equations for natural parameters of an ellipse. To illustrate the technique we used these equations in numerical examples with various location of measurements points on $\partial \Omega$. We also handled the problem in three dimensions where we proved uniqueness for an ellipsoid in some particular cases.