  We consider the identification of spatially distributed parameters under
$H^1$ regularization. Solving the associated minimization problem by
Gauss-Newton iteration results in linearized problems to be solved in each step
that can be cast as boundary value problems involving a low-rank modification
of the Laplacian. Using algebraic multigrid as a fast Laplace solver, the
Sherman-Morrison-Woodbury formula can be employed to construct a preconditioner
for these linear problems which exhibits excellent scaling w.r.t. the relevant
problem parameters. We first develop this approach in the functional setting,
thus obtaining a consistent methodology for selecting boundary conditions that
arise from the $H^1$ regularization. We then construct a method for solving the
discrete linear systems based on combining any fast Poisson solver with the
Woodbury formula. The efficacy of this method is then demonstrated with scaling
experiments. These are carried out for a common nonlinear parameter
identification problem arising in electrical resistivity tomography.
