Solving semi-linear elliptic optimal control problems with $$L^1$$-cost via regularization and RAS-preconditioned newton methods

Abstract We present a new parallel computational framework for the efficient solution of a class of $$\varvec{L}^{\varvec{2}}$$ / $$\varvec{L}^{\varvec{1}}$$ -regularized optimal control problems governed by semi-linear elliptic partial differential equations (PDEs). The main difficulty in solving this type of problem is the nonlinearity and non-smoothness of the $$\varvec{L}^{\varvec{1}}$$ -term in the cost functional, which we address by employing a combination of several tools. First, we approximate the non-differentiable projection operator appearing in the optimality system by an appropriately chosen regularized operator and establish convergence of the resulting system’s solutions. Second, we apply a continuation strategy to control the regularization parameter to improve the behavior of (damped) Newton methods. Third, we combine Newton’s method with a domain-decomposition-based nonlinear preconditioning, which improves its robustness properties and allows for parallelization. The efficiency of the proposed numerical framework is demonstrated by extensive numerical experiments.