Matthias Heinkenschloss - Research
Optimization of Time-Dependent Distributed Systems
Many systems can be modeled by linear or nonlinear time-dependent
partial differential equations (PDEs). Optimization of such systems, in the
context of optimal control, optimal design, or parameter estimation play
an important role in science and engineering. Algorithms for the
solution of time-dependent PDEs often involve marching in time, starting
from an initial condition. In optimization, however, the values of the
solution of the PDEs at later times feed into the
optimization at early times. This coupling in time makes the practical
solution of these very large-scale optimization problems challenging.
Direct application of generic large-scale optimization methods is
often infeasible because of excessive storage requirements.
To cope with the complexity of time-dependent PDE optimization
problems, storage management techniques (snapshot techniques)
have been introduced and suboptimal control schemes, such as reduced bases
techniques or instantaneous control, have been proposed.
For specific problems, reduced bases techniques or instantaneous control
have been applied successfully.
However, the analysis of these techniques is still incomplete and the limits
of their applicability are not clearly described.
The goal of this research is to improve the understanding
and the analysis of existing suboptimal control methods,
and to develop new methods for the fast solution of time-dependent
PDE optimization problems.
Applications that motivate our research and that serve as test
examples are specific optimal control problems in fluid
mechanics.
This research was supported by the National Science Foundation.