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

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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.