A regularization operator for source identification for elliptic PDEs

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