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：本文提出了一种高效的一阶原始对偶方法，用于求解非光滑偏微分方程约束优化问题。该方法的高效性源于无需在优化算法的每次迭代中求解偏微分方程或其线性化形式。相反，我们将优化算法与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）交织执行，每次优化迭代仅执行一步线性系统求解。控制参数根据优化算法的要求在每次迭代中更新。我们在二阶增长条件下证明了该方法的线性收敛性，并通过数值实验展示了其在涉及边界测量反问题的一系列偏微分方程模型上的计算性能。