We provide a global convergence proof of the recently proposed sequential homotopy method with an inexact Krylov--semismooth-Newton method employed as a local solver. The resulting method constitutes an active-set method in function space. After discretization, it allows for efficient application of Krylov-subspace methods. For a certain class of optimal control problems with PDE constraints, in which the control enters the Lagrangian only linearly, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method is faster than using direct linear algebra for the 2D benchmark and allows for the parallel solution of large 3D problems.
翻译:我们提供了一种全球趋同证明,证明最近提出的连续同质同质方法与不精确的Krylov-semismoth-Newton方法相匹配,该方法被用作本地求解器。由此得出的方法构成了功能空间中的一种主动设定方法。在分解后,它允许高效应用Krylov-Sub空间方法。对于某种因PDE限制而导致的最佳控制问题,即控制仅线性进入拉格朗江,我们提议和分析一种基于双舒尔补充方法的高效、平行、对称正确定的先决条件。我们得出的结果是,一个条件恶劣且高度非线性的基准优化问题,其结果为椭圆部分差异方程式和控制界限。所产生的方法比使用直线对2D基准的直线代数还快,并允许对大3D问题采取平行的解决办法。