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PDE-constrained optimization problems with control or state constraints arechallenging from an analytical as well as numerical perspective. Thecombination of these constraints with a sparsity-promoting $\rm L^1$ termwithin the objective function requires sophisticated optimization methods. Wepropose the use of an Interior Point scheme applied to a smoothed reformulationof the discretized problem, and illustrate that such a scheme exhibits robustperformance with respect to parameter changes. To increase the potency of thismethod we introduce fast and efficient preconditioners which enable us to solveproblems from a number of PDE applications in low iteration numbers and CPUtimes, even when the parameters involved are altered dramatically.

Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms