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

Efficient solution of parameter identification problems with $H^1$ regularization