Simultaneous Analysis and Design in Pde-constrained Optimization a Dissertation Submitted to the Institute for Computational and Mathematical Engineering and the Committee on Graduate Studies of Stanford University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
Simultaneous Analysis and Design in Pde-constrained Optimization a Dissertation Submitted to the Institute for Computational and Mathematical Engineering and the Committee on Graduate Studies of Stanford University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
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偏微分约束优化中的同步分析与设计部分满足哲学博士学位要求向斯坦福大学计算与数学工程研究所和研究生委员会提交的论文
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发表时间：
2013
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通讯作者：
Youngsoo Choi
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作者：
Youngsoo Choi
本文提出了求解某些类型偏微分方程组约束优化问题的新方法。所采取的方法是增强最先进的PDE方法。同时对偏微分方程变量和优化变量进行求解。这对偏微分方程组方法产生了影响，因为它的核心从求解一个非线性方程组转变为寻找一个非线性鞍点。这进而改变了在每次迭代中求解的线性方程的性质。在我们解决的问题中，目标函数必须与给定的目标状态相匹配。为了与目标匹配，同时考虑了体积和边界控制。将正则化添加到目标函数中，以帮助稳定性并便于计算算法中出现的线性系统的解。求解这类线性系统一直是许多研究的重点，提出了许多方法。如何针对特定系统中出现的偏微分方程组约束优化问题同时有效地进行求解，是我们工作的重点。新方法是通过修改尖端软件Aero-S来实现的。给出了各种问题的数值结果，包括扑翼问题、机器人控制问题和热控制问题。
New methods for solving certain types of PDE-constrained optimization problems are presented in this thesis. The approach taken is to augment state-of-the-art PDE methods. The PDE variables and the optimization variables are solved for simultaneously. This impacts the PDE method by changing the core from solving a system of nonlinear equations to that of finding a nonlinear saddle point. This in turn alters the character of the linear equations that are solved at each iteration. In the problem we address, the objective function has to match a given target state. Both volume and boundary controls are considered in order to match the target. Regularization is added to the objective function to aid stability and to facilitate computing the solution of the linear systems that arise within the algorithm. Solving such linear systems has been the focus of much research, with many methods being proposed. How to do this simultaneously and efficiently for the specific systems that arise in the PDE-constrained optimization problems of interest is the main focus of our work. The new methods have been implemented by modifying the cutting-edge software AERO-S. Numerical results are presented for a variety of problems including a flapping wing, a robotic control problem, and a thermal control problem.