Simultaneous Optimization with Unsteady Partial Differential Equations
Simultaneous Optimization with Unsteady Partial Differential Equations
复制标题
非定常偏微分方程联立优化
DOI：
10.18154/rwth-2017-06795
复制
发表时间：
2017
影响因子：
5.6
通讯作者：
M. Frank
中科院分区：
文献类型：
--
作者：
Stefanie Günther;N. Gauger;Qiqi Wang;M. Frank
非定常偏微分方程 (PDE) 的优化问题是应用数学中最具挑战性的领域之一。通常采用基于梯度的优化方案，其中通过设计参数的迭代更新来减少时间平均数量。伴随方法为梯度评估提供了强大的工具。然而，传统优化方法每一步中重复求解非定常动力学和伴随方程会带来显着的计算复杂性。当考虑混沌系统时，这个问题变得更具挑战性，因为基于伴随的梯度通常涉及时空边值问题的求解。为了减少传统优化方法的整体运行时间，这项工作的重点是将现有的非稳态 PDE 求解器集成到同步优化框架中。特别是，追求同步一次性方法。一次性优化算法将基于伴随的设计更新纳入到模拟底层偏微分方程的过程中，以实现最优性。由于常见的非定常仿真代码以向前时间推进的方式求解非定常动力学，在每个时间步迭代求解非线性方程，因此从仿真到一次性优化的转变并非易事。本文提出了实现这一目标的三种新颖方法。第一种方法将设计更新嵌入到一系列时间推进方案中，该方案在每个时间步均采用近似解决方案。在外部优化循环的每次迭代中，通过近似伴随和设计更新步骤来增强简化的时间推进方案，从而同时达到可行性和最优性。使用非稳态雷诺平均纳维-斯托克斯求解器演示了该方法在最优主动流量控制问题中的应用。与传统的优化方法相比，获得了三倍的加速系数。第二种方法涉及并行时间一次性优化方法。该方案利用迭代、非侵入式多重网格算法应用于非稳态时间推进方案的时域。然后在每次多重网格迭代后合并伴随和设计更新。时间并行一次性方法通过将计算工作负载沿时域分配给多个处理器来提高效率。该方法的潜力针对平流主导的模型问题得到了证明。与传统的时间序列方法相比，这里的加速因子明显提高了 24 倍。第三种方法解决了混沌偏微分方程的优化问题。设计了非稳态时间推进方案的重新表述，以实现向前和向后的时间信息传播。新公式能够通过调整初始条件来补偿设计的变化。由此产生的边值问题在时空中迭代解决，这样基于伴随的设计更新就可以自然地集成。致谢。首先，我要感谢我的导师Nicolas Gauger教授在过去五年里给予我的持续支持。他对我的科学事业和个人发展的周到指导使我建立了信任的合作关系。我感谢他努力创造最好的工作环境，提供频繁的参加外部会议的机会，并将我介绍给科学界的许多研究人员。此外，我感谢他对我在麻省理工学院 (MIT) 和阿贡国家实验室进行研究的鼓励和支持。我要感谢 Qiqi Wang 教授为本文提供的共同指导，以及在我访问麻省理工学院期间和之后进行的许多富有成效的讨论和编码会议。与他一起工作非常鼓舞人心，非常愉快。此外，我还要感谢 Martin Frank 教授修改了这篇论文并将我纳入他在亚琛 MathCCES 的工作组。我要感谢凯泽斯劳滕 SciComp 和亚琛 MathCCES 的所有同事，感谢他们是出色的同事，并营造了愉快的工作氛围。我特别要感谢 Armin Westerkamp 在无数的午餐和咖啡休息时间进行了愉快的讨论。我还要感谢蒂姆·阿尔布林 (Tim Albring) 在多次电话中的耐心和支持，以及马克斯·萨格鲍姆 (Max Sagebaum)、丽莎·库什 (Lisa Kusch) 和埃姆雷·厄兹卡亚 (Emre Özkaya)。感谢我的同事和朋友 Hossein Gorji、Muhammad Hassan、Florian Zwicke、Christian Jörres 和 Roland Siegbert 对本文的校对工作。我衷心感谢所有朋友在过去几年里给予我的爱和支持。特别是，我要感谢罗兰·西格伯特（Roland Siegbert）多次带我走出办公室，骑上自行车。我感谢他在本文最后阶段的耐心和帮助。最重要的是，我感谢我的家人。你们一直在我身边，相信我，支持我，激励我，激励我。没有你我就不可能完成这一切。混沌是自然法则；秩序是人类的梦想。
Optimization problems subject to unsteady partial differential equations (PDEs) comprise one of the most challenging areas of applied mathematics. Gradient-based optimization schemes are typically employed, where an time-averaged quantity is reduced by iterative updates of design parameters. The adjoint approach provides a powerful tool for gradient evaluation. Yet significant computational complexities arise from repeatedly solving the unsteady dynamics and the adjoint equation in each step of conventional optimization methods. The problem becomes even more challenging when chaotic systems are considered, as adjoint-based gradients often involve the solution of a space-time boundary value problem. In order to reduce the overall runtime of conventional optimization methods, this work focuses on integrating existing unsteady PDE solvers into a simultaneous optimization framework. In particular, the simultaneous One-shot approach is pursued. The One-shot optimization algorithm incorporates adjoint-based design updates towards optimality into the process of simulating the underlying PDEs. Since common unsteady simulation codes resolve the unsteady dynamics in a forward time-marching manner, solving nonlinear equations iteratively at each time step, the transition from simulation to One-shot optimization is non-trivial. Three novel approaches to achieve this are presented in this thesis. The first approach embeds design updates into a sequence of time-marching schemes that adopt approximate solutions at each time step. In each iteration of an outer optimization cycle, the reduced time-marching scheme is enhanced by approximate adjoint and design update steps, such that feasibility and optimality are reached simultaneously. An application of the method is demonstrated for an optimal active flow control problem using an unsteady Reynolds-averaged Navier-Stokes solver. A speedup factor of three is obtained in comparison to a conventional optimization method. The second approach is concerned with the parallel-in-time One-shot optimization method. This scheme utilizes an iterative, non-intrusive multigrid algorithm applied to the time domain of unsteady time-marching schemes. Adjoint and design updates are then incorporated after each multigrid iteration. The parallel-in-time One-shot method draws its efficiency from distributing computational workload to multiple processors along the time domain. The potential of the method is demonstrated for an advection-dominated model problem. Here, a significantly higher speedup factor of 24 is achieved in comparison to a conventional time-serial method. The third approach addresses optimization with chaotic PDEs. A reformulation of the unsteady time-marching scheme is devised that enables forwardand backward-in-time information propagation. The new formulation is able to compensate changes in the design by adjusting the initial conditions. The resulting boundary value problem is solved iteratively in space-time, such that adjoint-based design updates can be integrated naturally. Acknowledgements. Foremost, I want to thank my supervisor Prof. Nicolas Gauger for his constant support during the past five years. His thoughtful guidance concerning my scientific career and my personal development has emerged a trustful cooperation. I am thankful for his effort in creating the best-possible working environment, for providing the frequent opportunities to attend external conferences and introducing me to many researchers in the scientific community. Further, I am grateful for his encouragement and support of my research stays at the Massachusetts Institute of Technology (MIT) and the Argonne National Laboratory. I would like to thank Prof. Qiqi Wang for co-advising this thesis and for many fruitful discussions and coding sessions during and after my visits at MIT. It was very inspiring and a great pleasure to work with him. Further, I would like to thank Prof. Martin Frank for revising this thesis and integrating me into his working group at MathCCES in Aachen. I would like to thank all my colleagues at SciComp in Kaiserslautern and MathCCES in Aachen for being great coworkers and for creating a pleasant working atmosphere. In particular, I would like to thank Armin Westerkamp for enjoyable discussions during uncountable lunch and coffee breaks. I also thank Tim Albring for his patience and support during numerous phone calls, as well as Max Sagebaum, Lisa Kusch and Emre Özkaya. For proofreading this thesis, I would like to thank my colleagues and friends Hossein Gorji, Muhammad Hassan, Florian Zwicke, Christian Jörres and Roland Siegbert. I gratefully thank all my friends for their love and support during the last years. Especially, I would like to thank Roland Siegbert for bringing me out of the office and onto the bike for quite so many times. I am grateful for his patience and help during the final stages of this thesis. Most notably, I thank my family. You have always been there for me, believed in me, supported, motivated and inspired me. I would not have accomplished this without you. Chaos was the law of nature; Order was the dream of man.
具有不稳定偏微分方程的同步单步一次性优化
DOI：
10.1016/j.cam.2015.07.033
发表时间：
2016
期刊：
J. Comput. Appl. Math.
影响因子：
--
作者：
S. Günther;N.R. Gauger;Q. Wang
通讯作者：
Q. Wang
离散伴随求解器稳定性的递归投影法评估
DOI：
10.2514/6.2017-3664
发表时间：
2017
期刊：
影响因子：
--
作者：
T. Albring;T. Dick;N. R. Gauger
通讯作者：
N. R. Gauger