  The subject of this thesis is in the area of Applied Mathematics known as
Inverse Problems. Inverse problems are those where a set of measured data is
analysed in order to get as much information as possible on a model which is
assumed to represent a system in the real world. We study two inverse problems
in the fields of classical and quantum physics: QCD condensates from tau-decay
data and the inverse conductivity problem. We use a functional method which
allows us to extract within rather general assumptions phenomenological
parameters of QCD (the condensates) from a comparison of the time-like
experimental data with asymptotic space-like results from theory. The price to
be paid for the generality of assumptions is relatively large errors in the
values of the extracted parameters. Although we do not claim that our method is
superior to other approaches, we hope that our results lend additional
confidence to the numerical results obtained with the help of methods based on
QCD sum rules. In this thesis, also two approaches of EIT image reconstruction
are proposed. The first is based on reformulating the inverse problem in terms
of integral equations. This method uses only a single set of measurements for
the reconstruction. The second approach is an algorithm based on linearisation
which uses more then one set of measurements. A promising result is that one
can qualitatively reconstruct the conductivity inside the cross-section of a
human chest. Even though the human volunteer is neither two-dimensional nor
circular, such reconstructions can be useful in medical applications:
monitoring for lung problems such as accumulating fluid or a collapsed lung and
noninvasive monitoring of heart function and blood flow.
