On the Structure of Uniqueness in Linear Inverse Source Source Problems

Abstract

The formulation and solution of inverse problems require a functional analytic embedding in each case. As an important example for a linear inverse source problem, the inverse gravimetric problem is considered in the frame of Hilbert function space. Different ways are demonstrated how the uniqueness and non-uniqueness of the solvability of this inverse problem can be characterized. For this purpose, bases functions being linearly independent or orthogonal as well as trial functions, e. g. point masses, are efficient tools. They permit description of mass distributions with zero potential in the outer space and give the possibility to determine and include additional conditions which can lead to uniqueness to a certain degree if the solution belongs to a corresponding model class. Especially for the model class of finite point mass systems, the unique solvability of the inverse gravimetric problem is proved.