Mathematical Sciences: Optimization Methods for Optimal Control and Parameter Identification Problems

数学科学：最优控制和参数辨识问题的优化方法

• 批准号: 9403699
• 负责人: Matthias Heinkenschloss
• 金额: $ 5.7万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403699&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Optimization; Methods; Optimal

9403699 Heinkenschloss This research is concerned with the development and analysis of fast and robust methods for the solution of optimal control and parameter identification problems governed by nonlinear partial differential equations. These problems are usually formulated in infinite dimensional function spaces; a discretization of these problems leads to large sparse nonlinear programming problems. Many of these problems, in particular, parameter identification problems, have an inherent ill-posedness. These structures will be utilized in the design and analysis of the optimization methods. The algorithms operate on a sequence of nested discretizations. They use mesh-independence properties and grid refinements to compute good starting values in an efficient way. Moreover, they will incorporate multilevel ideas to obtain inexpensive and good derivative information from coarse grids. Iterative solvers will be used for the inexact solution of subproblems to address the large scale structure of the problems. To globalize the convergence and to address the ill-posedness, trust region strategies will be used. The algorithms will be formulated in a rather modular form allowing the incorporation of application dependent efficient solvers, such as domain decomposition methods and multigrid methods. The mathematical formulation of many problems in development and production leads to optimal control and parameter identification problems. Examples include the design of aircraft wings, the control of heating processes, and the nondestructive testing of materials. These mathematical models are complex and their numerical solution is very time consuming and often nearly impossible if 'of-the-shelf-methods' are used. The goal of this research project is the development of new, efficient and robust algorithms for the solution of these problems exploiting their inherent structure.

小行星9403699 本研究关注的是发展和分析的快速和强大的方法解决最优控制和参数识别问题的非线性偏微分方程。 这些问题通常是在无穷维函数空间中制定的;这些问题的离散化导致大型稀疏非线性规划问题。 这些问题，特别是参数识别问题，有一个固有的不适定性。 这些结构将被用于设计和分析的优化方法。 该算法对嵌套离散化序列进行操作。 他们使用网格独立属性和网格细化以有效的方式计算良好的初始值。 此外，他们将纳入多层次的想法，以获得廉价和良好的衍生信息，从粗网格。 迭代求解器将用于子问题的不精确解，以解决问题的大规模结构。 为了全球化的收敛和解决不适定性，信赖域策略将被使用。 该算法将制定在一个相当模块化的形式，允许将应用程序相关的高效求解器，如区域分解方法和多重网格方法。 开发和生产中的许多问题的数学公式导致最优控制和参数识别问题。 例子包括飞机机翼的设计，加热过程的控制，以及材料的无损检测。 这些数学模型是复杂的，并且它们的数值解是非常耗时的，并且如果使用“现成方法”，则通常几乎是不可能的。本研究项目的目标是开发新的，有效的和强大的算法，利用其固有的结构，这些问题的解决方案。