Local properties of the finite element solutions to PDE constrained optimal control problems

本课题的研究目的是为平流主导的最优控制问题的有限元解的局部性质奠定坚实的理论基础。众所周知，平流-扩散偏微分方程（PDEs）很难近似，因为它们的解经常表现为不连续、分层和激波。最优控制问题的解的结构更加复杂，因为在最优性系统中控制平流主导的偏微分方程与伴随平流主导的偏微分方程的耦合。众所周知，应用于单个平流主导的偏微分方程的有限元方法的收敛行为可能与应用于平流主导的最优控制问题的有限元方法的收敛行为有很大不同。了解全局和局部收敛行为对于可靠和有效地解决平流主导的最优控制问题至关重要，特别是在存在控制和状态约束以及依赖于点态信息的目标函数的情况下。该项目旨在加深我们对各种问题的理解，并帮助开发可靠的数值方法。数学被证明是非常有用的建模许多现实生活中的问题来自环境，技术，气候等。然而，许多数学模型需要不能直接测量的特殊参数。例子可以是建模技术设计中的形状，环境过程中的物理系数，导航中的控制等，需要进行估计。在数学上，对这些参数的估计通常会导致以系统偏微分方程（PDEs）形式存在约束的优化问题。通常，这种偏微分方程系统是很容易理解的，并且有许多可用的数值技术来求解它。然而，它并不立即适用于pde底层系统的方法也适用于约束优化问题。在我们之前的工作中，我们在一个简单的模型问题中展示了这种差异。在本提案中，我们打算研究涵盖更广泛应用的更复杂的模型问题。

The research objective of this project is to develop a strong theoretical foundation on local properties of finite element solutions for advection-dominated optimal control problems. The advection-diffusion partial differential equations (PDEs) are known to be difficult to approximate because their solutions often exhibit discontinuities, layers, and shocks. The structure of the solutions to optimal control problems is even more complicated because of the coupling in the optimality system of the governing advection-dominated PDE with an advection-dominated adjoint PDE. It is known that convergence behavior of finite element methods applied to single advection-dominated PDEs can be very different from the convergence behavior of finite element methods applied to advection-dominated optimal control problems. Understanding the global and local convergence behavior is crucial for reliable and efficient solution of advection-dominated optimal control problems, especially in the presence of control and state constraints, and objective functions that depend on pointwise state information. This project intends to deepen our understanding for various problems and to help develop reliable numerical methods. Mathematics proved to be extremely useful in modeling many real life problems coming from environment, technology, climate, and etc. However, many mathematical models require special parameters that can not be measured directly. Examples can be shapes in modeling technological devises, physical coefficients in environmental processes, controls in navigation and etc. and need to be estimated. Mathematically, estimation of such paramters often leads to optimization problems with constraints in the form of system partial differential equations (PDEs). Usually, such system of PDEs is well understood and there are many available numerical techniques to solve it. However, it does not immediately apply that the method that works well for the underlying system of PDEs will work for the constrained optimization problem. In our previous work, we showed such differences in the case of a simple model problem. In this proposal we intend to investigate more complicated model problems that cover a broader range of applications.