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

本研究的重点是发展和分析求解非线性偏微分方程控制下的最优控制和参数辨识问题的快速和鲁棒方法。这些问题通常在无限维函数空间中表述；这些问题的离散化导致了大型稀疏非线性规划问题。其中许多问题，特别是参数识别问题，具有固有的不适定性。这些结构将在设计和分析中应用优化方法。该算法在一系列嵌套离散化上运行。它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起始值。此外，它们将结合多层次的思想，从粗网格中获得廉价而良好的衍生信息。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。为了实现全局收敛和解决病态性问题，将采用信任域策略。算法将以相当模块化的形式制定，允许合并应用相关的高效求解器，例如域分解方法和多网格方法。开发和生产中许多问题的数学公式导致最优控制和参数辨识问题。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。这些数学模型非常复杂，它们的数值解非常耗时，如果使用“现成的方法”，通常几乎是不可能的。这个研究项目的目标是开发新的、高效的、鲁棒的算法来解决这些问题，利用它们的固有结构。

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.