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The numerical solution of optimization problems with partial differential equation (PDE) constraints is vital to a growing number of science and engineering applications. The development of robust and efficient algorithms for the solution of these optimization problems presents many challenges that arise out of, e.g., the intricate mathematical structure of these problems, the complicated interactions between numerical methods for PDE and optimization, the large-scale of the optimization problems, and the increasing complexity of applications. To identify and overcome these challenges an integrated approach is needed that builds on a variety of mathematical sub-disciplines, such as theory of PDEs, distributed parameter systems, numerical solution of PDEs, numerical optimization, and numerical linear algebra. This international workshop has brought together some of the leading experts in the fast developing field of optimization problems with PDE constraints to present recent developments in this area as well as to identify open problems and further research needs.
Among the themes of this workshop were the design and analysis of approaches for the solution of PDE constrained optimization problems with additional point-wise constraints on controls and states (the solution of the governing PDE). State constrained problems are particularly challenging because of the low regularity properties of the Lagrange multipliers associated with point-wise constraints on the states. A second theme was the development of adaptive methods for the solution of PDE constrained optimization problems and, more generally, the development of optimization level model reduction techniques for these problems. The goal here is to develop models (through, e.g., mesh adaptation or proper orthogonal model reduction) of the PDE constrained optimization problems that capture the relevant features of the optimization problems with a specified accuracy, but involve as few degrees of freedom as possible and, hence, are computationally less expensive to work with. A third theme was The efficient solution of linear systems arising in optimization algorithms for discretized PDE constrained optimization problems represented another theme. Finally, a number of talks presented advances and challenges in the solution of PDE constrained optimization problems arising in important science and industrial applications.
Cite this article
Ronald H.W. Hoppe, Matthias Heinkenschloss, Volker Schulz, Numerical Techniques for Optimization Problems with PDE Constraints. Oberwolfach Rep. 3 (2006), no. 1 pp. 585–652