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 high-standard 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.
  1. 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.
  2. Loewner evolution. The Loewner stochastic differential equation describes the evolution of a random process on the complex plane called the Schramm-Loewner 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 self-avoiding random walks, uniform spanning trees, and creek-crossing graphs.
  3. 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 deep-trap 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.
  4. 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.
  5. 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 rare-event estimation, via concepts such as the Kullback-Leibler cross-entropy, Shannon entropy and Radon-Nykodym derivatives. It requires a good knowledge of probability theory and mathematical statistics.
  6. Stochastic Processes on Exotic Spaces In this project you will explore how to simulate stochastic processes on non-standard 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.
  7. 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.
  8. Efficient Sampling in Bayesian Models. Many real-world statistical problems are nowadays solved via Bayesian models, where the aim is to sample from some posterior distribution. For high-dimensional problems this can be a very difficult or time-consuming 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 spin-off will be the development of software in these areas.
  9. Properties of the hit-and-run sampler. The hit-and-run sampler is one of the most successful methods for Markov chain Monte Carlo sampling. By using appropriate transformations of the data, the hit-and-run sampler can be restricted to a high-dimensional 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 high-dimensional 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 sub-problem, or carry out computer experiments, read research papers, etc. Moreover, you will be paid for your work!


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 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 non-Honours 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 research-only 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

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 Because of the all-pervasiveness of statistics, it is an extremely dynamic field of research. Many new ideas and breakthroughs are discovered each day.