We introduce an efficient first-order primal-dual method for the solution of nonsmooth PDE-constrained optimization problems. We achieve this efficiency through not solving the PDE or its linearisation on each iteration of the optimization method. Instead, we run the method interwoven with a simple conventional linear system solver (Jacobi, Gauss-Seidel, conjugate gradients), always taking only one step of the linear system solver for each step of the optimization method. The control parameter is updated on each iteration as determined by the optimization method. We prove linear convergence under a second-order growth condition, and numerically demonstrate the performance on a variety of PDEs related to inverse problems involving boundary measurements.

翻译：我们提出了一种高效的一阶原始-对偶方法，用于求解非光滑PDE约束优化问题。该方法的效率源于在优化方法的每次迭代中无需完全求解PDE或其线性化方程。相反，我们将该方法与简单的常规线性系统求解器（如雅可比迭代、高斯-赛德尔迭代、共轭梯度法）交织运行，每步优化方法仅执行一步线性系统求解器操作。控制参数按照优化方法确定的规则在每次迭代中更新。我们证明了在二阶增长条件下的线性收敛性，并通过涉及边界测量的反问题相关的多种PDE数值实验验证了该方法的性能。