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

在最优控制、最优设计或参数估计的背景下，线性或非线性抛物型微分方程的优化在科学和工程中起着重要的作用。求解抛物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，抛物方程的解在后期的值会在早期的优化中得到反馈。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具有挑战性。为了应对抛物线系统精确优化所需的存储和计算机时间需求，最近提出了所谓的次最优控制技术，如减基技术和瞬时控制。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描述。本研究将选择的次优控制技术集成到一个优化框架中，其中它们被解释为截断迭代方法或用作优化方法的前置条件。这提高了我们对这些技术的理论认识，拓宽了它们的适用性。所得到的方法适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，微型机电系统可以用来改变飞机机翼上的流动特性以减少阻力，或者可以调整加热器以达到炉内所需的温度分布，同时最大限度地减少能源消耗。许多系统可以用数学方程式来模拟。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的系统输入。对于可以通过适度复杂的数学方程充分建模的系统，这是常规和成功的。然而，其他系统的详细数学描述，包括飞机机翼上的气流或炉内的温度分布，是如此复杂，以至于目前用于确定最优控制策略的数学技术需要如此庞大的计算机资源，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，并通过流体力学中的选定应用证明这些工具的实用性。

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.