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

时相关偏微分方程控制的新型多重射击优化算法

• 批准号: 1819144
• 负责人: Matthias Heinkenschloss
• 金额: $ 31.92万
• 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: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1819144&HistoricalAwards=false
• 关键词: Novel; Multiple; Shooting; Algorithms; Optimization

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.

数学优化在工程系统的优化设计及其高效运行中起着至关重要的作用。例如，油藏管理需要将水注入井中，以将石油储量通过复杂的地质结构推至生产井，以实现收入最大化。此类系统通过瞬态偏微分方程 (PDE) 进行建模，并涉及许多设计或决策变量，例如需要确定的因井而异以及随时间变化的注入速率。这项研究为此类与时间相关的问题开发了新的数学优化算法。 具体来说，本研究旨在开发新算法，将直接多重射击（MS）公式有效地应用于由时间相关的偏微分方程控制的最优控制和最优设计问题，并展示这些算法在流量控制中的应用的性能。直接 MS 公式将基础偏微分方程分解为较短时间子区间上的方程，并将这些方程在时间区间边界处耦合。这些耦合条件必须在求解时得到满足，但在优化算法的迭代过程中则不然。通过子问题的卓越稳定性、求解算法的增强收敛性以及并行性的引入，可以利用这一点来实现此类问题数值求解的实质性改进。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，耦合条件是额外的约束。对于由（离散）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，已成功应用于常微分方程控制问题的 MS 公式的现有优化方法在 PDE 设置中实际上是不可行的。本研究的目标是 1) 将 MS 公式注入一阶梯度优化算法中，以扩展其对基础 PDE 解可能在数值上不稳定的问题的适用性，以及 2) 开发基于模型简化的新二阶迭代方法显着降低传统顺序二次规划方法中出现的大型二次规划求解的计算成本。将提供新方法的收敛分析，并将在流量控制中的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智力优点和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Towards Data-Driven Model Reduction of the Navier-Stokes Equations using the Loewner Framework

使用 Loewner 框架实现纳维-斯托克斯方程的数据驱动模型简化

• DOI: 10.1007/978-3-030-90727-3_14
• 发表时间: 2022-04
• 期刊: Active Flow and Combustion Control 2021
• 作者: Diaz, Alejandro N.;Heinkenschloss, Matthias
• 通讯作者: Heinkenschloss, Matthias

A new unified model to simulate columns with multiple phase divisions and their impact on energy savings

模拟多相塔塔的新统一模型及其对节能的影响

• DOI: 10.1016/j.compchemeng.2020.106937
• 发表时间: 2020-09-02
• 期刊: Comput. Chem. Eng.
• 作者: L. C. Biasi;M. Heinkenschloss;F. R. Batista;R. Zemp;A. L. Romano;Antonio J. A. Meirelles
• 通讯作者: Antonio J. A. Meirelles

Reduced Order Model Hessian Approximations in Newton Methods for Optimal Control

最优控制牛顿法中的降阶模型 Hessian 近似

• DOI: 10.1007/978-3-030-95157-3_18
• 发表时间: 2022-07
• 期刊: Realization and Model Reduction of Dynamical Systems - A Festschrift in Honor of the 70th Birthday of Thanos Antoulas
• 作者: Heinkenschloss, Matthias;Magruder, Caleb
• 通讯作者: Magruder, Caleb

Parastillation and metastillation applied to bioethanol and neutral alcohol purification with energy savings

平行蒸馏和转移蒸馏应用于生物乙醇和中性醇的节能纯化

• DOI: 10.1016/j.cep.2021.108334
• 发表时间: 2021-05
• 期刊: Chemical Engineering and Processing - Process Intensification
• 作者: Biasi, Lilian C.K.;Batista, Fabio R.M.;Zemp, Roger J.;Romano, Ana L.R.;Heinkenschloss, Matthias;Meirelles, Antonio J.A.
• 通讯作者: Meirelles, Antonio J.A.

Efficient solution of large-scale algebraic Riccati equations associated with index-2 DAEs via the inexact low-rank Newton-ADI method

通过不精确的低秩 Newton-ADI 方法有效求解与指数 2 DAE 相关的大规模代数 Riccati 方程

• DOI: 10.1016/j.apnum.2019.11.016
• 发表时间: 2018-04-04
• 期刊: Applied Numerical Mathematics
• 影响因子: 2.8
• 作者: P. Benner;M. Heinkenschloss;J. Saak;H. Weichelt
• 通讯作者: H. Weichelt