Dirk Kroese's Research Projects
On this page I have listed a number of possible research projects for
students who want to do a PhD, Honours, Masters or a Vacation
Scholarship with me. If you are interested in doing research under my
supervision, please drop by my office, send me an email, and/or
talk to my past and present PhD and Honours students.
My Research
My research lies, broadly, in the fields of Monte Carlo statistical methods,
random algorithms and stochastic modelling; in short, anything that deals with
the theory and application of randomness.
I have published widely in highstandard
international journals (over 90 publications as of 01/06/13) and
written four books with the top publishers in the field (Wiley and Springer).
Many
of the research projects below will be
related to Monte Carlo methods.
Doing research with me would bring
you to the forefront of research in this area. This would mean that
you would be actively cooperating with a group of experts around the
world (e.g. at the Technion, University of Amsterdam, University of
Twente, Stanford, MIT, University of Aarhus)
and would be doing relevant (i.e. publishable!) research. Moreover,
because these methods are relatively new, there are still a lot of things to be discovered.
Publications are nowadays essential in order to obtain scholarships, etc.
Research Topics
Each topic below can be researched at different levels, from Summer Vacation
Scholarship to PhD. It is important to note that the projects below
are only small sample of possible research directions. I very much
welcome your input with regard to what you would like to investigate.

Rare event simulation for spatial stochastic processes.
Spatial processes are mathematical models for spatial data; that is,
spatially arranged measurements and patterns. Spatial processes come
in many different forms and shapes, ranging from point processes to
Markov random fields. The availability of fast computers and
advances in Monte Carlo simulation methods have greatly enhanced the
understanding of spatial processes. However, the analysis of spatial
processes under extreme conditions is still not well understood. The purpose of this project is to
develop new theory and applications for the efficient simulation of
spatial processes conditioned on rare events.

Loewner evolution.
The Loewner stochastic differential equation describes the evolution
of a random process on the complex plane called the SchrammLoewner
evolution (SLE).
It has been found recently that important
random processes on the plane are in the limit described by SLE processes.
This project is about further exploring the connections between complex
analysis and geometrical random objects such as
selfavoiding random walks, uniform spanning trees, and
creekcrossing graphs.

Random walks in random environments.
A random walks in a random environment is an extention of the ordinary
random walk. The idea is that the random environment greatly
influences the progress of the random walker. The behaviour of such
processes is of much theoretical interest, and is closely related to
percolation and deeptrap phenomena in statistical physics.
Applications are found for example in charge transport processes.
The purpose of this project is to better understand the theoretical
properties of random walks in random environments by exploring a range
of models for the random environment in onedimensional and
multidimensional settings. Efficient simulation techniques
for such processes play an important role in the analysis.

Counting via Monte Carlo. One can use Monte Carlo algorithms
to solve difficult
counting problems! For example, how many paths are there in a graph,
or
how many truth assignments are there for a certain SAT problem? Using
random algorithms to calculate deterministic quantities may seem
at first counter intuitive, but recent development have shown that
this approach yields substantial improvements on deterministic
methods. For some counting questions deterministic approaches are not
even feasible.
The goal of this project is to develop simple but clever
randomized
algorithms for a wide range of counting objects. An additional
advantage of the randomized approach is that complexity results may be
obtained. This project has
high output potential for students who are confident in both
probability and discrete mathematics.
 Information and Large Deviations in Statistical Estimation. Large deviations theory
describes how random processes behave away from their usual
regime. In this project you will investigate the role that
Information Theory and Large Deviations play in the theory of
rareevent
estimation, via concepts such as the
KullbackLeibler crossentropy, Shannon entropy and RadonNykodym
derivatives. It requires a good knowledge of probability
theory and mathematical statistics.
 Stochastic Processes on Exotic Spaces
In this project you will explore how to simulate stochastic processes
on nonstandard spaces. For example, how does one generate a Wiener
process (Browian motion) on a sphere or torus? One area of application
is the numerical solution of partial
differential equations. This would be a good project for someone
interested in both simulation and mathematical analysis.

CE Methods for Optimal Control. Optimal control problems, e.g., as arising in
epidemic control, engineering, and finance, can often be
formulated as functional optimisation problems, where the objective is
to find an optimal continuous "control" function. In this project you will
investigate how the CE method and its offspring can be used to locate the optimal
control function. Some familiarity with optimisation techniques is required.

Efficient Sampling in Bayesian Models. Many realworld
statistical problems are nowadays solved via Bayesian models, where
the aim is to sample from some posterior distribution. For
highdimensional problems this can be a very difficult or timeconsuming task. The
purpose of this project is to investigate and
develop efficient Monte Carlo
sampling procedures, such as the recently discovered generalised
splitting method.
Applications range from computational biology and medicine to financial
engineering and risk analysis; a spinoff will be the development of software in these areas.

Properties of the hitandrun sampler.
The hitandrun sampler is one of the most successful methods for
Markov chain Monte Carlo sampling. By using appropriate
transformations of the data, the hitandrun sampler
can be restricted to a highdimensional unit hypercube. This opens the
way to deal with many complicated estimation problems in a
simple and uniform way. In this project you will investigate the usefulness and
limitations of this approach for highdimensional problems,
and find new theory and efficient algorithm to deal with a range of
concrete applications.
Why Research?
Why should you consider doing research?
Research is the driving force behind our universities.
Without research our knowledge would become "stale" and "bookish".
Although some students, like most Hobbits, only like to hear what is already familiar, others will be delighted to discover a whole
unexplored world of knowledge by doing research. The University
of Queensland prides itself in being one of the leading Research
Universities of the country. The School of Mathematics and Physics
is one of the "power houses"
of research within the University, as evidenced by the large number of
competitive research grants and international journal papers.
Summer Vacation Scholarship
The first opportunity to do real research in mathematics/statistics
is the Summer Vacation Scholarship. The summer break is the time
where most academics are busy writing research papers. As a summer
vacation scholar you will be able to slot into this research
and you may be asked to work on a not too difficult subproblem,
or carry out computer experiments,
read research papers, etc. Moreover, you will be paid for your work!
Honours
The second opportunity to do research occurs in the Honours year. Many people believe
that the Honours year is merely a precursor to a PhD. This is not
true. The purpose of the Honours year is to
 give you a lot more
knowledge about the area(s) of your interest  there is simply not
enough time to cover everything in the undergraduate curriculum
 to teach you how to conduct research.
Although many students
continue with a postgraduate degree (Masters or PhD) after their
Honours degree, a significant group of students goes straight into
industry, business or government, where their honours degree, and
their proven ability to do research, gives them a distinct advantage
over nonHonours candidates.
Postgraduate Studies
Then there are the Postgraduate studies. The Coursework Masters degree
is similar in nature to (but 1 semester longer than) an Honours degree.
Here you
do 1 year of coursework and 2 half semesters (or 1 full
semester) of research. You can do this in stages (Postgraduate
Certificate > Postgraduate Diploma > Masters). The Master of Philosophy (MPhil)
is a 1 1/2 to 2 years researchonly programme (Honours is
required). Finally, the ultimate research degree is the PhD,
which takes 3 1/2 4 years and requires an Honours, GradDipl or
Masters degree.
Research in Mathematics and Statistics
Mathematics is the
language of science and Statistics provides the mathematical language
and techniques necessary for understanding and dealing with chance and
uncertainty in Nature. Mathematics and Statistics are essential in ANY field of science.
Without it we would not be able to make much scientific progress,
simply because we would not be able to communicate our findings in a
clear and precise way. Have you ever been frustrated by the more
"indefinite" disciplines where for each question there are 10 equally
plausible answers (opinions)? In mathematics/statistics we are in the fortunate situation where there is only One Truth,
and you can help to discover it.
Statistics involves the design, collection, generation, analysis and
interpretation of numerical data, with the aim of extracting patterns
and other useful information. Examples include
 the analysis of DNA and
protein sequences,
 the construction of evolutionary trees from genetic
data,
 the improvement of medical treatments via experimental designs,
 the assessment of drought conditions through meteorological
data, and

the implementation of random algorithms for optimisation and
estimation.
A main feature of statistics is the development and use of
statistical and probabilistic models for random phenomena, which can
be analysed and used to make principled predictions and
decisions. Examples of such models can be found in
 biology (genetics,
population modelling),
 finance (stock market fluctuations, insurance
claims),
 physics (quantum mechanics/computing),
 medicine
(epidemiology, spread of HIV/AIDS),
 telecommunications (internet
traffic, mobile phone calls),
and
 reliability (safety of oil rigs,
aircraft failure), to name but a few.
Because of the
allpervasiveness of statistics, it is an extremely dynamic field of
research. Many new ideas and breakthroughs are discovered each day.