Automated Extension of Fixed Point PDE Solvers for Optimal Design with Bounded Retardation
Automated Extension of Fixed Point PDE Solvers for Optimal Design with Bounded Retardation
复制标题
用于有限延迟优化设计的定点 PDE 求解器的自动扩展
DOI：
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发表时间：
2012
期刊：
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通讯作者：
T. Slawig
中科院分区：
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作者：
N. Gauger;A. Griewank;A. Hamdi;Claudia Kratzenstein;E. Özkaya;T. Slawig
研究了用伪时间步进或不动点迭代求解状态方程的pde约束优化问题。我们提出了一种在每个迭代步骤中同时提高原始、对偶可行性和最优性的技术，从而实现了状态和伴随迭代的耦合以及控制/设计更新。我们的目标是获得这个耦合迭代相对于原始状态的有界延迟，因为后者在许多情况下只有一个接近1的q因子。为此目的，基于双增广拉格朗日，它可以被证明是一个精确的惩罚函数，我们详细讨论了适当的控制或设计空间前置条件的选择，讨论了实现问题，并给出了收敛分析。我们给出了数值例子，其中包括流体力学中的形状设计和气候模型中的参数优化的应用。
We study PDE-constrained optimization problems where the state equation is solved by a pseudo-time stepping or fixed point iteration. We present a technique that improves primal, dual feasibility and optimality simultaneously in each iteration step, thus coupling state and adjoint iteration and control/design update. Our goal is to obtain bounded retardation of this coupled iteration compared to the original one for the state, since the latter in many cases has only a Q-factor close to one. For this purpose and based on a doubly augmented Lagrangian, which can be shown to be an exact penalty function, we discuss in detail the choice of an appropriate control or design space preconditioner, discuss implementation issues and present a convergence analysis. We show numerical examples, among them applications from shape design in fluid mechanics and parameter optimization in a climate model.