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

平流扩散方程最优控制问题的间断伽辽金法

• 批准号: 0811167
• 负责人: Dmitriy Leykekhman
• 金额: $ 10.68万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0811167&HistoricalAwards=false
• 关键词: Discontinuous; Galerkin; Methods; Optimal; Control

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.

有几种方法可以解决现实生活中的问题。 一种有吸引力且相对便宜的方法是设计一种数学模型，该模型描述了某些物理现象，并使用此模型进行预测。 为了验证模型，重要的是，预测与实际观察结果一致。 通常，数学模型具有未知或不确定的几个参数。 考虑到一些观察结果，可以调整这些参数，以减少预测值和观察到的值之间的差异。 换句话说，通过完善模型，我们希望最大程度地减少该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佳控制问题。 在此提案中，我们对偏微分方程（PDE）系统给出的约束的最佳控制问题感兴趣。 特别是，我们对描述流动演变的模型感兴趣。 此类问题的相应PDE称为对流扩散方程。 这些方程是基本的，对此类方程的解决方案通常表现出“非平滑行为”，例如冲击，边界和内部层以及界面不连续性。 这种现象经常在现实中观察到，并具有严重的计算和分析挑战，以便以数字解决此类问题。 在设计一种数值方法时，重要的是要知道该方法在这种不连续性附近的行为，以及所产生的效果是全球还是局部。多年来，已经设计了许多竞争方法。 在此提案中，我们打算结合一个不连续的Galerkin（DG）方法。 这些方法最近受到了很多关注。 主要吸引力的特征是，DG方法使用不连续的函数近似未知的解决方案，并且原则上非常适合功能值急剧变化的问题。在此建议中，我们打算开发新的分析工具，使我们能够在最佳控制问题的背景下分析此类方法。 我们还计划在计算上与其他常用的地球物理流动问题相比，与其他常用的方法相比，DG方法估计重要物理量的优势，例如底部阻力系数和涡流粘度。