Interior-point methods and preconditioning for PDE-constrained optimization problems involving sparsity terms

Partial differential equation (PDE)–constrained optimization problems with control or state constraints are challenging from an analytical and numerical perspective. The combination of these constraints with a sparsity-promoting L1 term within the objective function requires sophisticated optimization methods. We propose the use of an interior-point scheme applied to a smoothed reformulation of the discretized problem and illustrate that such a scheme exhibits robust performance with respect to parameter changes. To increase the potency of this method, we introduce fast and efficient preconditioners that enable us to solve problems from a number of PDE applications in low iteration numbers and CPU times, even when the parameters involved are altered dramatically.