Optimization of Parabolic Systems: Iterative Methods, Suboptimal Controls, and Preconditioning

抛物线系统的优化：迭代方法、次优控制和预处理

• 批准号: 0075731
• 负责人: Matthias Heinkenschloss
• 金额: $ 14万
• 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: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0075731&HistoricalAwards=false
• 关键词: Optimization; Parabolic; Systems; Iterative; Methods

Optimization of linear or nonlinear parabolic differential equationsin the context of optimal control, optimal design, or parameter estimation plays an important role in science and engineering. Algorithms for thesolution of parabolic equations often involve marching in time, startingfrom an initial condition. In optimization, however, the values of thesolution of the parabolic equation at later times feed into theoptimization at early times. This coupling in time makes the practicalsolution of these very large-scale optimization problems challenging.To cope with storage and computer time demands required by an exactoptimization of parabolic systems, so-called suboptimal control techniques, such as reduced bases techniques and instantaneous control have recently been proposed. The analysis of these techniques is still incomplete and the limits of their applicability are not clearly described. Thisresearch integrates selected suboptimal control techniques into anoptimization framework, where they are interpreted as truncated iterative methods or are used as preconditioners in optimization methods.This improves our theoretical understanding of these techniques and broadens their applicability. The resulting methods are applied to specific optimal control problems in or related to fluid mechanics. Optimal control attempts to determine system parameters or inputs to increase the performance of the system. For example, micro electromechanicalsystems may be used to alter the flow characteristics on an aircraft wing to reduce drag, or heaters may be adjusted to achieve a desired temperatureprofile in a furnace while minimizing energy consumption. Many systemscan be modeled by mathematical equations. In this case mathematical techniques can be used, at least in principle, to determine the optimalsystem inputs. For systems that can be adequately modeled by moderatelycomplex mathematical equations this is done routinely and successfully.Detailed mathematical descriptions of other systems, including flow over an aircraft wing or the temperature distribution in a furnace, however,are so complex that present mathematical techniques for the determinationof optimal control strategies require such large computer resources thatrender them impractical. The goal of this research is to develop and analyzecomputational mathematics tools for the determination of optimal control strategies for a class of complex systems and the demonstration ofthe practicability of these tools using selected applications in fluid mechanics.

最优控制、最优设计或参数估计中的线性或非线性抛物型微分方程的优化在科学和工程中发挥着重要作用。抛物线方程的求解算法通常涉及从初始条件开始的时间推进。然而，在优化中，稍后抛物线方程的解的值会反馈到早期的优化中。这种时间上的耦合使得这些非常大规模的优化问题的实际解决方案具有挑战性。为了满足抛物线系统的精确优化所需的存储和计算机时间需求，最近出现了所谓的次优控制技术，例如减少基数技术和瞬时控制被提议。对这些技术的分析仍然不完整，并且其适用性的限制也没有清楚地描述。这项研究将选定的次优控制技术集成到优化框架中，将它们解释为截断迭代方法或用作优化方法中的预处理器。这提高了我们对这些技术的理论理解并拓宽了它们的适用性。由此产生的方法应用于流体力学中或与流体力学相关的特定最优控制问题。 最优控制试图确定系统参数或输入以提高系统的性能。例如，微机电系统可用于改变飞机机翼上的流动特性以减少阻力，或者可调节加热器以在熔炉中实现所需的温度分布，同时最小化能量消耗。许多系统可以通过数学方程来建模。在这种情况下，至少在原则上可以使用数学技术来确定最佳系统输入。对于可以通过适度复杂的数学方程进行充分建模的系统，这是常规且成功的完成。然而，其他系统的详细数学描述（包括飞机机翼上的流动或熔炉中的温度分布）非常复杂，因此需要提出数学技术确定最优控制策略需要大量的计算机资源，这使得它们不切实际。本研究的目标是开发和分析计算数学工具，用于确定一类复杂系统的最佳控制策略，并使用流体力学中的选定应用来演示这些工具的实用性。