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有效运行中起着至关重要的作用。例如，储油层的管理需要注入例如，将水注入威尔斯井，通过复杂的地质结构将石油储量推到生产威尔斯井，目的是使收入最大化。类似这样的系统由时间相关偏微分方程（PDE）建模，并且涉及许多设计或决策变量，例如需要确定的在威尔斯之间和随时间变化的注入速率。本研究开发了新的数学优化算法，这样的时间依赖的问题。 具体而言，本研究的目的是开发新的算法，有效地应用直接多次射击（MS）配方的最优控制和最优设计问题的时间相关的偏微分方程，并演示了这些算法的性能，在流量控制中的应用。直接MS配方分解成方程的基础偏微分方程的时间较短的子间隔和耦合这些在时间间隔的边界。这些耦合条件必须在解时满足，而不是在优化算法的迭代期间。利用这一点，通过子问题的上级稳定性，增强的收敛性能的解决方案的算法，并引入并行性，以实现这些问题的数值解的实质性改善。然而，MS公式有一个代价：时间间隔边界处的辅助初始数据是额外的优化变量，耦合条件是额外的约束。对于由（离散化）偏微分方程控制的问题，这会导致优化变量和约束的数量大幅增加。由于这些增加，现有的优化方法，已成功地应用到MS配方的常微分方程的问题，实际上是不可行的PDE设置。本研究的目标是：1）将MS公式注入到一阶梯度优化算法中，以扩展其适用性，以解决潜在PDE的解可能在数值上不稳定的问题; 2）开发基于模型降阶的新的二阶迭代方法，以大幅降低求解传统序列二次规划方法中出现的大型二次规划的计算成本。将提供新方法的收敛性分析，并将在流量控制应用中演示这些方法。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.