论文标题
BV（扩展版）中用于PDE受限最佳控制的路径遵循的不偏用牛顿方法
A path-following inexact Newton method for PDE-constrained optimal control in BV (extended version)
论文作者
论文摘要
我们研究了PDE受限的最佳控制问题，该问题涉及有界变异作为控制的功能，并包括目标中控制的电视符号。我们将纽顿牛顿的路径遵循方法应用于平滑电视节目和添加$ H^1 $正则化所带来的问题。我们在无限维度的环境中证明，首先，这些辅助问题的解决方案融合到了原始问题的解决方案，其次，当将控制的牛顿牛顿牛顿方法不精确地融合到最佳系统中，该方法将其用于最佳系统的重新印度，其中控制状态作为邻接状态的隐含功能。我们显示有限元近似的收敛性，提供了一种全球化的预处理不精源的牛顿方法作为离散辅助问题的求解器，并将其嵌入到不精确的路径遵循方案中。我们使用完全显式解决方案构建了二维测试问题，并提出数值结果，以说明该方法的准确性和鲁棒性。 这是“计算优化和应用程序”中出现的相应期刊文章的扩展版本。它包含日记版本中省略的一些证明。
We study a PDE-constrained optimal control problem that involves functions of bounded variation as controls and includes the TV seminorm of the control in the objective. We apply a path-following inexact Newton method to the problems that arise from smoothing the TV seminorm and adding an $H^1$ regularization. We prove in an infinite-dimensional setting that, first, the solutions of these auxiliary problems converge to the solution of the original problem and, second, that an inexact Newton method enjoys fast local convergence when applied to a reformulation of the optimality system in which the control appears as implicit function of the adjoint state. We show convergence of a Finite Element approximation, provide a globalized preconditioned inexact Newton method as solver for the discretized auxiliary problems, and embed it into an inexact path-following scheme. We construct a two-dimensional test problem with fully explicit solution and present numerical results to illustrate the accuracy and robustness of the approach. This is an extended version of the corresponding journal article appearing in "Computational Optimization and Applications". It contains some proofs that are omitted in the journal's version.
