  In this paper we present a methodology that allows the efficient computation
of the topological derivative for semilinear elliptic problems within the
averaged adjoint Lagrangian framework. The generality of our approach should
also allow the extension to evolutionary problems and other nonlinear problems.
Our strategy relies on a rescaled differential quotient of the averaged adjoint
state variable which we show converges weakly to a function satisfying an
equation defined in the whole space. A unique feature and advantage of this
framework is that we only need to work with weakly converging subsequences of
the differential quotient. This allows the computation of the topological
sensitivity within a simple functional analytic framework under mild
assumptions.
