Novel Finite Element Methods for Elliptic Distributed Optimal Control Problems

椭圆分布最优控制问题的新颖有限元方法

• 批准号: 1913035
• 负责人: Susanne Brenner
• 金额: $ 26.23万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1913035&HistoricalAwards=false
• 关键词: Novel; Finite; Element; Methods; Elliptic

Optimal control problems with elliptic partial differential equation constraints appear in many optimal design processes in engineering and science. In these problems the state (output) is connected to the control (input) through an elliptic partial differential equation, and the objective is to find the control that will produce a desired state in an optimal fashion. The proposed research is on the design, analysis and efficient implementation of novel numerical methods for such problems, with applications to mechanical engineering, electrical engineering and materials science.Traditional numerical approaches for these optimal control problems treat the control as the primary unknown. The resulting finite element methods only involve low order elements. The convergence analysis, where the error estimates for the control, the state and the adjoint state are intertwined, is substantially more complicated than the convergence analysis for elliptic boundary value problems. In contrast, the approach in the proposed research treats the state as the primary unknown by reformulating the optimal control problems as variational inequalities for the state. A new analytical framework developed recently by the PI and the Co-PI shows that the convergence analysis for these elliptic variational inequalities can be obtained by using the same tools for the convergence analysis for elliptic boundary value problems. Consequently many finite element methods originally intended for elliptic boundary value problems can also be applied to the optimal control problems constrained by elliptic partial differential equations. The goal of the proposed research is to apply this new insight to design novel finite element methods for optimal control problems with general cost functionals, problems with semi-linear second order and fourth order elliptic partial differential equation constraints, problems for electromagnetics and problems with rough coefficients that appear in materials science.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.

椭圆型偏微分方程约束的最优控制问题出现在工程和科学中的许多优化设计过程中。 在这些问题中，状态（输出）通过椭圆偏微分方程与控制（输入）相连，目标是找到以最佳方式产生所需状态的控制。 本论文的主要研究内容是设计、分析和有效实现新的数值方法来求解此类问题，并将其应用于机械工程、电气工程和材料科学等领域。所得到的有限元方法只涉及低阶单元。 收敛性分析，其中的控制，状态和伴随状态的误差估计交织在一起，是远远比椭圆边值问题的收敛性分析复杂。相比之下，在所提出的研究方法将状态作为主要的未知数，通过重新制定的最优控制问题的状态变分不等式。PI和Co-PI最近提出的一个新的分析框架表明，这些椭圆变分不等式的收敛性分析可以通过使用椭圆边值问题收敛性分析的相同工具来获得. 因此，许多原来用于椭圆边值问题的有限元方法也可以应用于椭圆型偏微分方程约束的最优控制问题。本研究的目标是将这一新的认识应用于设计新的有限元方法来求解具有一般代价泛函的最优控制问题，具有半线性二阶和四阶椭圆型偏微分方程约束的问题，该奖项反映了NSF的法定使命，并被认为是值得支持的，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。

项目成果

Adaptive C0 interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients

具有 Cordes 系数的 Hamilton-Jacobi-Bellman 方程的自适应 C0 内罚方法

• DOI: 10.1016/j.cam.2020.113241
• 发表时间: 2021
• 期刊: Journal of Computational and Applied Mathematics
• 影响因子: 2.4
• 作者: Brenner, Susanne C.;Kawecki, Ellya L.
• 通讯作者: Kawecki, Ellya L.

Additive Schwarz Preconditioners for ?0 Interior Penalty Methods for a State Constrained Elliptic Distributed Optimal Control Problem

状态约束椭圆分布式最优控制问题的 ?0 内罚方法的加性 Schwarz 预条件子

• 发表时间: 2023
• 期刊: Springer
• 作者: Brenner, Susanne C.;Sung, Li-yeng;Wang, Kening
• 通讯作者: Wang, Kening

A cubic C0 interior penalty method for elliptic distributed optimal control problems with pointwise state and control constraints

• DOI: 10.1016/j.rinam.2020.100119
• 发表时间: 2020-08
• 期刊: Results in Applied Mathematics
• 影响因子: 2
• 作者: S. C. Brenner;L. Sung;Zhiyu Tan

A General Superapproximation Result

一般的超近似结果

• DOI: 10.1515/cmam-2020-0120
• 发表时间: 2020
• 期刊: Computational Methods in Applied Mathematics
• 影响因子: 1.3
• 作者: Brenner, Susanne C.

Finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state

• DOI: 10.1007/s11081-020-09491-1
• 发表时间: 2020-01
• 期刊: Optimization and Engineering
• 影响因子: 2.1
• 作者: S. C. Brenner;L. Sung;W. Wollner