  We consider a bilevel optimization approach in function space for the choice
of spatially dependent regularization parameters in TV image restoration
models. First- and second-order optimality conditions for the bilevel problem
are studied, when the spatially-dependent parameter belongs to the Sobolev
space $H^1(\Omega)$. A combined Schwarz domain decomposition-semismooth Newton
method is proposed for the solution of the full optimality system and local
superlinear convergence of the semismooth Newton method is analyzed. Exhaustive
numerical computations are finally carried out to show the suitability of the
approach.
