  We study whether a modified version of Tikhonov regularization can be used to
identify several local sources from Dirichlet boundary data for a prototypical
elliptic PDE. This paper extends the results presented in [5]. It turns out
that the possibility of distinguishing between two, or more, sources depends on
the smoothing properties of a second or fourth order PDE. Consequently, the
geometry of the involved domain, as well as the position of the sources
relative to the boundary of this domain, determines the identifiability.
  We also present a uniqueness result for the identification of a single local
source. This result is derived in terms of an abstract operator framework and
is therefore not only applicable to the model problem studied in this paper.
Our schemes yield quadratic optimization problems and can thus be solved with
standard software tools. In addition to a theoretical investigation, this paper
also contains several numerical experiments.
