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

9403699 Heinkenschoss这项研究致力于开发和分析快速和稳健的方法来解决由非线性偏微分方程组控制的最优控制和参数辨识问题。这些问题通常是在无限维函数空间中描述的；这些问题的离散化导致了大型稀疏非线性规划问题。这些问题中的许多问题，特别是参数辨识问题，具有内在的不适性。这些结构将用于优化方法的设计和分析。这些算法对一系列嵌套的离散化进行操作。它们使用与网格无关的属性和网格细化来高效地计算良好的起始值。此外，它们将结合多层次的想法，从粗略的网格中获得廉价和良好的派生信息。对于子问题的不精确解，将使用迭代求解器，以解决问题的大规模结构。为了使趋同全球化并解决不适当性，将使用信任区战略。算法将以相当模块化的形式制定，允许结合依赖于应用的高效求解器，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表述导致了最优控制和参数辨识问题。例如，飞机机翼的设计，加热过程的控制，以及材料的无损检测。这些数学模型很复杂，它们的数值解非常耗时，如果使用现成的方法，往往几乎不可能。这个研究项目的目标是开发新的、高效的和健壮的算法来解决这些问题，利用它们的内在结构。

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.