论文标题
基于steklov-Poincaré-type指标，用于PDE约束形状优化的非线性共轭梯度方法
Nonlinear Conjugate Gradient Methods for PDE Constrained Shape Optimization Based on Steklov-Poincaré-Type Metrics
论文作者
论文摘要
近年来，基于形状微积分的形状优化引起了很多关注，尤其是在有效优化算法的开发，分析和修改方面。在本文中，我们提出并研究了基于steklov-Poincaré-type指标的非线性共轭梯度方法，以解决由部分微分方程约束的形状优化问题的解决方案。我们将这些方法嵌入了基于梯度的形状优化方法的一般算法框架中，并讨论了算法的数值离散化。我们将提出的非线性偶联梯度方法与已经建立的梯度下降和有限的内存BFGS方法进行数值比较，以在几个基准问题上进行形状优化。结果表明，所提出的非线性共轭梯度方法在实践中表现良好，并且它们是已经建立的基于梯度的形状优化算法的有效且有吸引力的补充。
Shape optimization based on shape calculus has received a lot of attention in recent years, particularly regarding the development, analysis, and modification of efficient optimization algorithms. In this paper we propose and investigate nonlinear conjugate gradient methods based on Steklov-Poincaré-type metrics for the solution of shape optimization problems constrained by partial differential equations. We embed these methods into a general algorithmic framework for gradient-based shape optimization methods and discuss the numerical discretization of the algorithms. We numerically compare the proposed nonlinear conjugate gradient methods to the already established gradient descent and limited memory BFGS methods for shape optimization on several benchmark problems. The results show that the proposed nonlinear conjugate gradient methods perform well in practice and that they are an efficient and attractive addition to already established gradient-based shape optimization algorithms.
