Novel Finite Element Methods for Elliptic Distributed Optimal Control Problems

椭圆型偏微分方程约束下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连接到控制（输入），目标是找到将以最优方式产生所需状态的控制。本研究旨在设计、分析和有效地实施新的数值方法来解决这些问题，并将其应用于机械工程、电气工程和材料科学。这些最优控制问题的传统数值方法将控制作为主要未知数。所得的有限元方法只涉及低阶单元。由于控制、状态和伴随状态的误差估计交织在一起，其收敛性分析比椭圆边值问题的收敛性分析要复杂得多。相反，本文提出的方法通过将最优控制问题重新表述为状态的变分不等式，将状态视为主要未知数。最近由PI和Co-PI提出的一种新的分析框架表明，这些椭圆型变分不等式的收敛性分析可以用与椭圆型边值问题收敛性分析相同的工具得到。因此，许多原先用于求解椭圆型边值问题的有限元方法也可以应用于求解椭圆型偏微分方程约束下的最优控制问题。提出的研究目标是将这种新的见解应用于设计新的有限元方法，以解决具有一般成本泛函的最优控制问题，具有半线性二阶和四阶椭圆偏微分方程约束的问题，电磁学问题以及材料科学中出现的粗糙系数问题。该奖项反映了美国国家科学基金会的法定使命，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。

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.