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Published online: 2006-12-31
Numerical Techniques for Optimization Problems with PDE Constraints
Matthias Heinkenschloss[1], Ronald H.W. Hoppe[2] and Volker Schulz[3] (1) Rice University, Houston, United States(2) Universität Augsburg, Germany
(3) Universität Trier, Germany
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.
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Heinkenschloss Matthias, Hoppe Ronald, Schulz Volker: Numerical Techniques for Optimization Problems with PDE Constraints. Oberwolfach Rep. 3 (2006), 585-652. doi: 10.4171/OWR/2006/11