Efficient Solution of Advection Dominated PDE Constrained Optimization Problems

平流主导偏微分方程约束优化问题的高效求解

• 批准号: 0915238
• 负责人: Matthias Heinkenschloss
• 金额: $ 26.48万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0915238&HistoricalAwards=false
• 关键词: Efficient; Solution; Advection; Dominated; PDE

The aim of this proposal is to develop, analyze and implement optimizationalgorithms that integrate multilevel iterative solvers, adaptive mesh refinement methods, and so-called `all-at-once' methods for the solution of optimization problems governed by advection dominated partial differential equations (PDEs). The presence of strong advection in the governing PDE creates many challenges forthe numerical solution of these optimization problems, beyond the challenges alreadyencountered in the numerical simulation of single advection dominated PDEs and beyondthe many difficulties in solving PDE constrained optimization problems with weak or no advection. One reason for the additional challenges arising in the optimization context is the presence of the so-called adjoint equation, which is also an advection dominated PDE, but with advection given by the negative of the advection in the governing equation. This can cause significant and perhaps unexpected differences in the behavior of discretization schemes and iterative solvers when they are extended from the application to single advection dominated PDEs to the solution of PDE constrained optimization problems. This research will analyze the sources of these differences, their impact on the quality of computed solutionand on the efficiency of solution algorithms. Another goal is to devise modifications of multilevel iterative solvers, adaptive mesh refinement methods, and so-called KKT solvers tomake them robust against the presence of advection.Many important real-life applications such as the shape optimization of technological devices, the optimal control of systems, and the identification of parameters in environmentalprocesses lead to optimization problems governed by systems of advection dominatedpartial differential equations. This research addresses mathematical and algorithmicissues that are crucial for the reliable and efficient solution of these problems. It will leadto a better theoretical understanding of the solution properties of these optimization problems as well as to new computational tools for their reliable and efficient solution. Furthermore, it will provide training opportunities for students in an important area of computational science.

该提案的目的是开发、分析和实现集成多级迭代求解器、自适应网格细化方法和所谓的“一次性”方法的优化算法，用于解决由平流主导的偏微分方程控制的优化问题（偏微分方程）。主导偏微分方程中强对流的存在给这些优化问题的数值求解带来了许多挑战，超出了单一对流主导偏微分方程数值模拟中已经遇到的挑战，也超出了求解弱对流或无对流的偏微分方程约束优化问题的许多困难。优化环境中出现额外挑战的原因之一是所谓的伴随方程的存在，该方程也是平流主导的偏微分方程，但平流由控制方程中平流的负值给出。当离散化方案和迭代求解器从应用扩展到单个平流主导的 PDE 到 PDE 约束优化问题的解决方案时，这可能会导致离散化方案和迭代求解器的行为出现显着的甚至意外的差异。 本研究将分析这些差异的来源、它们对计算解的质量和求解算法的效率的影响。另一个目标是设计多级迭代求解器、自适应网格细化方法和所谓的 KKT 求解器的修改，以使它们对平流的存在具有鲁棒性。许多重要的现实应用，例如技术设备的形状优化、系统，以及环境过程中参数的识别导致由平流主导偏微分方程系统控制的优化问题。这项研究解决了对于可靠有效地解决这些问题至关重要的数学和算法问题。它将带来对这些优化问题的解决方案属性的更好的理论理解，以及新的计算工具来提供可靠和有效的解决方案。此外，它将为学生提供计算科学重要领域的培训机会。