We present a new parallel computational framework for the efficient solution of a class of $L^2$/$L^1$-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 $L^1$-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.

翻译：本文提出了一种新的并行计算框架，用于高效求解一类由半线性椭圆偏微分方程（PDEs）控制的$L^2$/$L^1$正则化最优控制问题。求解此类问题的主要困难在于代价泛函中$L^1$项的非线性与非光滑性。我们通过结合多种工具来解决这一难题。首先，我们用一个适当选取的正则化算子来逼近最优性系统中出现的不可微投影算子，并证明了所得系统解的收敛性。其次，我们采用延拓策略来控制正则化参数，以改善（阻尼）牛顿法的性能。第三，我们将牛顿法与基于区域分解的非线性预条件技术相结合，这增强了算法的鲁棒性并实现了并行化。大量数值实验验证了所提数值框架的有效性。