@PhDThesis{ itlic:2000-011,
author = {Bharath Bhikkaji},
title = {Model Reduction for Diffusion Systems},
school = {Department of Information Technology, Uppsala University},
department = {Division of Systems and Control},
year = {2000},
number = {2000-011},
type = {Licentiate thesis},
month = dec,
abstract = {Diffusion phenomena has been studied with a lot of
interest, for a long time, due to its historical and
practical significance. In the recent days it has thrown a
lot of interest among control engineers, as more and more
practical systems, varying from stock markets to
environmental pollution, have been observed to involve
diffusion.
Diffusion systems are normally modeled by linear partial
differential equations (LPDE's) of the form \[
\frac{\partial T(x,t)}{\partial x} = \mathcal{L}T(x,t)
\label{eq1.0} \] where $ \mathcal{L}$ is a second order
linear spatial differential operator and $ T(x,t) $ is the
physical quantity, whose variations in the spatial domain
cause diffusion. To characterise diffusion phenomena, one
has to obtain the solution of (\ref{eq1.0}) either
analytically or numerically. Note that, since (\ref{eq1.0})
involves a second order spatial operator and a first order
time derivative, one needs at least two boundary conditions
in the spatial domain, $x$, and an initial condition at
time $t=0$, for determining $ T(x,t) $.
LPDE's of the type (\ref{eq1.0}) can be interpreted as
infinite order linear time invariant systems (LTI systems)
with inputs as boundary conditions. To compute the solution
of (\ref{eq1.0}) numerically, one has to approximate,
explicitly or implicitly, the underlying infinite order
system by a finite order system. Any numerical scheme,
which computes the solution of (\ref{eq1.0}), essentially
approximates the underlying infinite order LTI system by a
finite order LTI system. The efficiency of the
approximation, for a given problem, varies for the
different numerical schemes.
In this thesis, we make an attempt to explore more about
diffusion systems in general. As a starting point, we
consider a simple case of one dimensional heat diffusion
across a homogeneous region. The resulting LPDE is first
shown explicitly to be an infinite order dynamical system.
An approximate solution is computed from a finite order
approximation of the true infinite order dynamical system.
In this thesis, we first construct the finite order
approximations using certain standard PDE solvers based on
Chebyshev polynomials. From these finite order
approximations we choose the best one, from a model
reduction perspective, and use it as a benchmark model. We
later construct two more approximate models, by exploiting
the given structure of the problem and we show by
simulations that these models perform better than the
chosen benchmark. }
}