  We study a source identification problem for a prototypical elliptic PDE from
Dirichlet boundary data. This problem is ill-posed, and the involved forward
operator has a significant nullspace. Standard Tikhonov regularization yields
solutions which approach the minimum $L^2$-norm least-squares solution as the
regularization parameter tends to zero. We show that this approach 'always'
suggests that the unknown local source is very close to the boundary of the
domain of the PDE, regardless of the position of the true local source.
  We propose an alternative regularization procedure, realized in terms of a
novel regularization operator, which is better suited for identifying local
sources positioned anywhere in the domain of the PDE. Our approach is motivated
by the classical theory for Tikhonov regularization and yields a standard
quadratic optimization problem. Since the new methodology is derived for an
abstract operator equation, it can be applied to many other source
identification problems. This paper contains several numerical experiments and
an analysis of the new methodology.
