Theory and Numerics of Model Reduction
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Most numerical simulations are based on complex mathematical models, often
described by partial differential equations (PDEs).
A typical use of such simulations is the measurement and
control of output quantities such as heat, noise, and stress
at critical points of the domain with respect to a selected
set of input parameters. The fundamental idea of model reduction
is that this input-output behaviour can often be well approximated
by a much simpler model than needed for describing the entire
state of the simulation. In this lecture, we consider
automatic model reduction techniques that are primarily based on numerical
control theory. In contrast to classical approaches, such techniques
require little or no understanding of the underlying model.
Once model reduction has been performed, the original model can be
replaced by the resulting simpler model, leading to reduced simulation
times and greatly facilitating the further analysis and design of a
control system. For instance, often only a low-order model allows for
the use of more sophisticated robust and optimal control techniques.
With the advances of modern control theory, model reduction has become
an important and rapidely changing field with a large diversity of
application areas, including structural and fluid dynamics, biosystems,
circuit simulation,
and micro-electro-mechanical systems. |
Oberwolfach model reduction benchmark collection |
Lecturers
Martin Gutknecht
Daniel Kressner
Assistant: Christine Tobler
Dates
The lecture take place every Wednesday, 10-12, HG D 5.2. The first lecture is on
18.02.2009.
The exercises take place every Friday, 13-14, HG D 3.2.
The first exercise is on 20.02. and provides a brief introduction into the Matlab Control Toolbox (exact details to be provided soon).
Contents of this course
- Short introduction into the basics of control theory (state space formulation, transfer function, system norms, stability, controllability, observability, minimal realization).
- Balanced truncation model reduction
- H2-optimal model reduction
- POD
- Mathematical tools (rational interpolation, nonlinear approximation)
- Numerical techniques (ADI, Krylov subspace methods, sign function iteration)
- Selection of current research directions in model reduction
(nonlinear systems, descriptor systems, passivity preservation, structure preservation).
Teaching material
- Lectures 1 and 2: Introduction into mathematical control theory
(intro.pdf).
- Supplementary material for balanced truncation (balanced.pdf)
- Supplementary material for Lyapunov equations (lyapunov.pdf)
- Supplementary material for ADI methods (adi.pdf)
- T. Stykel: Model reduction for differential-algebraic equations (descriptor.pdf)
Exercises
See separate web site.
Literature
The lecture will be as self-contained as possible. Some additional literature on
control theory and model reduction:
Overview web pages on model reduction
modelreduction.com
Model reduction at Rice
Model reduction at MIT
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