Proximal Methods for Elliptic Optimal Control Problems with Sparsity Cost Functional

First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.