Novel Multiple-Shooting Algorithms for Optimization Governed by Time-Dependent Partial Differential Equations

数学优化在工程系统的优化设计和高效运行中起着至关重要的作用。例如，对油层的管理需要向油井注入水，以便将石油储量通过复杂的地质结构推向生产井，以实现收入最大化。这类系统由依赖时间的偏微分方程组(PDE)建模，涉及许多设计或决策变量，例如需要确定的不同井和不同时间的注入量。这项研究为这类时间相关问题开发了新的数学优化算法。具体地说，本研究旨在开发新的算法，将直接多重打靶(MS)公式有效地应用于时间相关偏微分方程组的最优控制和最优设计问题，并展示这些算法在流量控制应用中的性能。直接MS公式将基本的偏微分方程组分解成较短时间子间隔上的方程，并在时间间隔边界上耦合这些方程。这些耦合条件必须在解中满足，但在优化算法的迭代过程中不满足。利用这一点，通过子问题优越的稳定性，增强解算法的收敛特性，以及引入并行性，实现了对这类问题的数值解的实质性改进。然而，MS公式是有代价的：时间间隔边界处的辅助初始数据是额外的优化变量，而耦合条件是额外的约束。对于由(离散化的)PDE管理的问题，这会导致优化变量和约束的数量大幅增加。由于这些改进，已经成功地应用于由常微分方程组控制的MS公式的现有优化方法在偏微分方程组中实际上是不可行的。本研究的目标是：1)将MS公式引入一阶梯度优化算法，使其适用于偏微分方程解可能在数值上不稳定的问题；2)发展新的基于模型降阶的二阶迭代方法，以显著降低传统序列二次规划方法中求解大型二次规划的计算成本。将提供新方法的收敛分析，并将在流量控制应用程序中演示这些方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Mathematical optimization plays a crucial role in the optimal design of engineering systems and in their efficient operation. For example, management of oil reservoirs requires the injection of, e.g., water into wells to push oil reserves through complex geological structures to production wells with the goal to maximize revenue. Systems like this are modeled by time-dependent partial differential equations (PDEs) and involve many design or decision variables, such as injection rates that vary among wells and over time, that need to be determined. This research develops new mathematical optimization algorithms for such time-dependent problems. Specifically, this research aims to develop new algorithms for the efficient application of direct multiple shooting (MS) formulations to optimal control and optimal design problems governed by time dependent PDEs, and demonstrates the performance of these algorithms to applications in flow control. Direct MS formulations decompose the underlying PDEs into equations on shorter time subintervals and couple these at the time interval boundaries. These coupling conditions must be satisfied at the solution, but not during the iteration of an optimization algorithm. This is exploited to achieve substantial improvements in the numerical solution of such problems through superior stability properties of sub-problems, enhanced convergence properties of solution algorithms, and introduction of parallelism. However, MS formulations have a price: The auxiliary initial data at time interval boundaries are additional optimization variables and the coupling conditions are additional constraints. For problems governed by (discretized) PDEs this leads to huge increases in the number of optimization variables and constraints. Because of these increases, existing optimization approaches that have been successfully applied to MS formulations of problems governed by ordinary differential equations are practically infeasible in the PDE setting. The goals of this research are to 1) inject MS formulations into first-order gradient optimization algorithms to expand their applicability to problems where the solution of the underlying PDE may be numerically unstable and 2) develop new second-order iterative methods based on model reduction to substantially reduce the computational cost of solving large quadratic-programs that arise in conventional sequential quadratic programming approaches. Convergence analysis of the new methods will be provided and these methods will be demonstrated on applications in flow control.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.