A nonsmooth primal-dual method with interwoven PDE constraint solver
提出一种高效一阶原对偶方法求解非光滑PDE约束优化问题，通过交织简单线性系统求解器（如Jacobi、Gauss-Seidel、共轭梯度）避免每次迭代求解PDE，证明线性收敛并数值验证了在边界测量反问题中的性能。
Abstract 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.