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

本计画的研究目标是发展对流占优最佳控制问题之有限元素解的局部性质的坚实理论基础。对流-扩散偏微分方程（PDE）是一类难以近似的问题，因为其解往往存在间断、分层和激波。最优控制问题的解的结构甚至更加复杂，因为在最优性系统中的控制对流主导的PDE与对流主导的伴随PDE的耦合。众所周知，应用于单个对流占优偏微分方程的有限元方法的收敛行为可以与应用于对流占优最优控制问题的有限元方法的收敛行为非常不同。了解全局和局部收敛行为是至关重要的对流主导的最优控制问题的可靠和有效的解决方案，特别是在存在控制和状态约束，以及依赖于逐点状态信息的目标函数。本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。 数学在模拟环境、技术、气候等真实的生活问题中被证明是非常有用的。然而，许多数学模型需要特殊的参数，这些参数不能直接测量。例子可以是造型技术设计中的形状，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，这些参数的估计往往会导致系统偏微分方程（PDE）形式的约束优化问题。通常，这样的系统的偏微分方程是很好地理解，有许多可用的数值技术来解决它。然而，它并不立即适用的方法，适用于底层系统的偏微分方程将工作的约束优化问题。 在我们以前的工作中，我们在一个简单的模型问题的情况下显示了这种差异。在这个建议中，我们打算调查更复杂的模型问题，涵盖更广泛的应用。

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.