Stephen Joe

Contact details

Room: G.2.28
Email address:
Phone number: +64-7-838 4073
Postal address: Department of Mathematics
School of Computing and Mathematical Sciences
University of Waikato
Private Bag 3105
Hamilton 3240
New Zealand

Teaching in 2023

His 2023 teaching in Trimester A consisted of the second half of ENGEN301-23A (HAM) Engineering Mathematics and Modelling 3 as well as all of MATHS165-23A (HAM) General Mathematics and MATHS166-23A (HAM) Management Mathematics. MATHS165 and MATHS166 are taught together. His Trimester B teaching consists of COMPX367-23B (HAM) Computational Mathematics and COMPX567-23B (HAM) Advanced Computational Mathematics. COMPX367 and COMPX567 share some common classes.


Current research is on lattice methods for the numerical evaluation of integrals in many variables. He has also recently supervised two PhD students on statistical modelling.

He has been working on projects to do with the apparent success of quasi-Monte Carlo methods when applied to high-dimensional integrals arising in applications such as mathematical finance. Though the standard theory of complexity indicates that such problems are `intractable', there is no doubt that such integrals are being evaluated successfully with only hundreds or thousands of function evaluations required.

One way of explaining this success is to make use of weighted function spaces in which the variables are of differing importance. Earlier joint work with his (former) Ph.D. student, Professor Frances Kuo and Professor Ian Sloan of the University of New South Wales has shown that there do exist lattice rules which are `good' in this situation. Moreover, their generating vectors may be obtained by using a component-by-component construction.

Work on topics related to the construction of lattice rules is continuing. In particular, the results above are based on a L2 version of a weighted discrepancy. These results have been extended by him and a (former) PhD student, Vasile Sinescu, to a L version of a weighted discrepancy, sometimes referred to as a weighted star discrepancy.

He is also working on the structure of lattice rules.

Besides lattice rules, he and Frances Kuo have also done some work on Sobol' points, which are another type of quasi-Monte Carlo method. More details about this project are available at this link.

Brief history

Born and bred in Palmerston North, it was natural that Stephen should attend the university there. At Massey University he did a BSc(Hons) degree in statistics followed by a MSc in mathematics. While a student at Massey he developed an interest in numerical analysis. This interest led him to obtain a Commonwealth Scholarship which enabled him to do a PhD at the University of New South Wales (UNSW) in Sydney. His PhD research topic was on the numerical solution of integral equations (these are equations in which the unknown function occurs as part of an integral).

After completing his Ph.D., he worked at UNSW for five years (as a lecturer and as a postdoctoral research fellow) before taking up a lectureship at the University of Waikato in 1992. He was promoted to Senior Lecturer in 1995 and to Associate Professor in 2005. He was Associate Dean in the School of Computing and Mathematical Sciences from mid-2003 until the end of 2004. He was Chairperson of the Department of Mathematics for a three year term which ended at the end of 2007. He was then the Acting Dean in the School of Computing and Mathematical Sciences from 1 February to 30 September 2008. After that, he was the Deputy Dean of the Faculty of Computing and Mathematical Sciences from 1 October 2008 to 31 January 2019. He was the Acting Head of the School of Computing and Mathematical Sciences until 31 January 2020.

In 2011, he was a recipient of an award for Administrative Excellence and also received the 2011 Vice-Chancellor's Medal for Staff Excellence.

His interest in numerical analysis continues to this day. As mentioned above, his current research is on lattice methods for the numerical evaluation of integrals in many variables. Though numerical analysis may be considered part of applied mathematics, it is an area of research which makes extensive use of pure mathematics as well as computers. For instance, his research on lattice methods has made use of number theory and group theory and has involved the development of algorithms and computer programs to test the theory and implementation of the methods.