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

线性或非线性抛物型微分方程的最优化在最优控制、最优设计或参数估计等领域中起着重要的作用。求解抛物型方程的算法通常涉及从初始条件开始的时间推进。然而，在优化中，抛物方程的解的值在较晚的时候被馈送到早期的优化中。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。为了科普精确优化抛物系统所需的存储和计算机时间需求，最近提出了所谓的次优控制技术，如减少基技术和瞬时控制。对这些技术的分析仍然不完整，其适用性的限制也没有明确说明。本研究将所选择的次优控制技术整合到一个优化框架中，将其解释为截断迭代方法或作为优化方法中的预条件子，这提高了我们对这些技术的理论理解，拓宽了它们的适用范围。所得到的方法被应用到特定的最优控制问题或相关的流体力学。 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，或者加热器可被调节以在炉中实现期望的温度分布，同时最小化能量消耗。许多系统可以用数学方程来模拟。在这种情况下，至少在原则上，可以使用数学技术来确定最优系统输入。对于那些可以用较为复杂的数学方程充分模拟的系统，这是常规而成功地完成的。然而，对其他系统的详细数学描述，包括飞机机翼上的流动或炉内的温度分布，是如此复杂，以至于目前用于确定最佳控制策略的数学技术需要如此庞大的计算机资源，这使它们变得不切实际。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优控制策略，并使用选定的流体力学应用程序来演示这些工具的实用性。

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.