Discontinuous Galerkin Methods for Optimal Control Problems Governed by Advection-Diffusion Equations

有几种方法可以解决现实生活中的问题。一个有吸引力且相对便宜的方法是设计一个描述某些物理现象的数学模型，并用这个模型进行预测。为了验证模型，重要的是预测与实际观测相符。通常数学模型有几个未知或不确定的参数。给定一些观测值，可以调整这些参数，以减少预测值和观测值之间的差异。换句话说，通过改进模型，我们希望最小化模型产生的某些客观数量。从数学上讲，这些类型的问题可以归类为最优控制问题。在这个提议中，我们感兴趣的是由偏微分方程（PDEs）系统给出约束的最优控制问题。特别地，我们对描述流的演化的模型感兴趣。这类问题的相应偏微分方程称为平流扩散方程。这些方程是基本的，这些方程的解通常表现出“非光滑行为”，如激波、边界层和内层以及界面不连续。这种现象在现实中经常观察到，并且为了在数值上解决这类问题，具有严重的计算和分析挑战。在设计数值方法时，了解该方法在这些不连续点附近的表现，以及所得到的影响是全局的还是局部的，是非常重要的。多年来，人们设计了许多有竞争力的方法。在这个提议中，我们打算纳入一个不连续伽辽金（DG）方法族。这些方法最近受到了很多关注。主要的吸引人的特点是DG方法使用不连续函数来近似未知解，并且原则上非常适合于函数值急剧变化的问题。在本提案中，我们打算开发新的分析工具，使我们能够在最优控制问题的背景下分析这些方法。我们还计划在计算上证明DG方法在估计重要物理量（如底部阻力系数和涡流粘度）方面的优势，而不是其他常用的现实地球物理流动问题方法。

There are several methods to approach real-life problems. An attractive and relatively cheap approach is to design a mathematical model that describes some physical phenomena and to use this model to make predictions. To validate the model it is important that the predictions agree with real observations. Usually mathematical models have several parameters that are unknown or uncertain. Given some observations, these parameters can be tuned in order to reduce the discrepancy between the predicted and the observed values. In other words, by refining a model we want to minimize certain objective quantities produced by the model. Mathematically these types of problems can be classified as optimal control problems. In this proposal we are interested in optimal control problems with constraints given by systems of partial differential equations (PDEs). In particular, we are interested in models that describe an evolution of a flow. The corresponding PDEs for such problems are called advection-diffusion equations. These equations are fundamental and solutions to such equations often exhibit "nonsmooth behavior", like shocks, boundary and interior layers, and interface discontinuities. Such phenomena are often observed in reality and possess serious computational and analytical challenges in order to solve such problems numerically. In designing a numerical method it is very important to know how the method behaves in the neighborhood of such discontinuities, and whether or not the resulting effects are global or local. Over the years many competitive methods have been designed. In this proposal we intend to incorporate a family of Discontinuous Galerkin (DG) methods. These methods have received a lot of attention lately. The main attractive feature is that DG methods use discontinuous functions to approximate the unknown solutions and in principle are well suited for problems with sharp changes in function values.In this proposal we intend to develop new analytical tools that will enable us to analyze such methods in the context of optimal control problems. We also plan to demonstrate computationally the advantages of the DG methods for estimating important physical quantities, such as bottom drag coefficient and eddy viscosity, over other commonly used methods for real-life geophysical flow problems.