Total immersion in the field of applied mathematics

To some, ‘Schrödinger’ and ‘Faddeeva’ are mere surnames. To Stellenbosch University (SU) distinguished professor André Weideman, they are two of the core mathematical terms that he has worked with during his 40-year career in applied mathematics.

This specific equation and function have led to some of his most cited research papers on numerical analysis and scientific computing, and contributed to him receiving his first A1-rating from the National Research Foundation in 2022.

Weideman joined SU’s Department of Mathematical Sciences in 1999, after spending a number of fruitful, stimulating years in the 1990s as an associate professor at Oregon State University in the USA. He returned to South Africa with his wife, Marì, and three children to be closer to family. (As the son of a mathematics teacher from the Free State, Weideman is still a firm supporter of the Cheetahs rugby team.)

He is the only person from Africa thus far to be elected as a fellow of the prestigious international Society for Industrial and Applied Mathematics (SIAM). Upon his election in 2017, he was commended for his ability to produce “powerful and elegant numerical algorithms derived from complex analysis.”

He received a medal for research distinction from the South African Mathematical Society in 2019, and, in 2020, the SU Chancellor’s Award for Research. A year later, he was promoted to the level of distinguished professor.

When opportunity knocks

Applied mathematics, Weideman explains, is about solving mathematical problems that arise in real-world contexts in the fields of, for instance, engineering, finance and medicine. Here, solutions must be fast and accurate when implemented on a computer, and allow for minimal machine-generated rounding errors.

Many of Weideman’s applied maths colleagues work side by side with people in the industry to solve engineering- or production-related issues. Weideman himself works on more general problem classes rather than specific ones, hoping that he can thereby help others to do their jobs better.

His work has proved valuable to scientists and engineers working in the fields of fluid dynamics, spectroscopy, astrophysics and laser optics.

“I don’t choose my research fields; they choose me,” he notes. In a sense, he says, he’s a ‘when-opportunity-knocks’ type of person: He often finds problems to solve in others’ journal articles.

“When I see a mathematical equation or formula, I’ll start thinking about how to compute it efficiently. Only after working on it, might I realise how and where the solution can be applied.

“Of course, for every paper published, there are a thousand problems that I couldn’t solve,” he notes dryly. “You may think you have a brilliant idea, but if you cannot demonstrate that it’s better than the existing ones, it ends up in the wastepaper basket.”

Research highlights

Most of Weideman’s papers appear in top journals in the fields of applied mathematics, numerical analysis and scientific computing. He has also done work on algorithm and software development. Many of his codes, available online, are written in the programming languages MATLAB and Python.

A 1994 paper of his on the computation of the complex error function (also known as the ‘Fadeeva’ or ‘plasma dispersion function’) has become the standard against which all new methods for the computation of this function are compared.

Moreover, one of Weideman’s papers from 2000 on spectral differentiation matrices, available online along with a software package DMSUITE, has attracted just under 1 000 citations on Google Scholar thus far. Reviewers, who have given it a five-star rating on the MathWorks site, describe it as “intuitive, well documented and extremely useful.”

Weideman is an associate editor for two international journals, Electronic Transactions on Numerical Analysis and Numerical Algorithms, and was, until recently, an editorial board member for South Africa’s Quaestiones Mathematicae. He also reviews papers for some of the most prestigious journals in his field, as well as book proposals for SIAM and Springer.

For the last two decades, Weideman has served on the main organising committee of SANUM, the South African Numerical and Applied Mathematics Symposium. “This has grown to be a truly international meeting, with many leading international speakers.”

Some of his recent research highlights include co-organising a four-month research programme on complex analysis at the prestigious Isaac Newton Institute in Cambridge in the UK, and being an invited speaker at the 2019 meeting of the International Council for Industrial and Applied Mathematics (ICIAM) in Spain. The latter attracted more than 4 000 participants, and is the biggest meeting of its kind in the fields of industrial and applied mathematics.

How it all began

Weideman first started considering a career in mathematics in his matric year, 1975, after participating in the South African Mathematics Olympiad at Hentie Cilliers High in Virginia in the Free State.

“Somewhere in my office, I still have the book prize I received. It made me realise that I could try maths – without yet really knowing what applied mathematics is.”

During his BSc studies at the University of the Free State in the late 1970s, the laborious system of punch cards was still the norm in computer programming. He completed his master’s degree in 1980, using a mainframe computer.

He bought his first computer, an Apple IIe with a 64 KB memory system and a floppy drive, around 1983, before starting work on his PhD. His supervisor, Prof Ben Herbst, would later become a colleague at SU.

The Apple still has pride of place in his office, along with other computer paraphernalia that hold stories about his research career.

Total immersion

“Being a true researcher demands total immersion,” says Weideman.

He has carried this idea with him since 1986 when, as a newly capped PhD graduate, he spent a formative year at the Massachusetts Institute of Technology (MIT) in the USA.

At MIT, he met one of his lifelong mentors, Prof Nick Trefethen, with whom he still collaborates.

During the COVID-19-related lockdown in 2020, they spent hours on Zoom with Prof Yuji Nakatsukasa from Oxford University, discussing ideas about exponential node clustering in various rational approximation methods.

“Our eventual paper, published in Numerische Mathematik, formed a prominent part of Prof Trefethen’s prizewinning 2020 John von Neumann lecture, one of SIAM’s highest accolades. My contribution to it was acknowledged.”

A paper that Weideman and Trefethen published in 2014 in SIAM Review, the society’s flagship journal, has a long history. This article on the exponential convergence of the trapezoidal rule was triggered by one Weideman published in The American Mathematical Monthly in 2002.

“During my sabbatical in Oxford in 2004 and 2005, Trefethen suggested we turn this into a bigger paper. We did this, off and on, over the next eight years. It culminated in the 74-page behemoth published in 2014. Even though it can be categorised as a review paper, it contains a lot of new material and previously unnoticed connections between different areas of computational mathematics.”

Weideman is proud of the legwork that went into getting the necessary history citations in the paper just right.

“Not sparing any costs or time, we even visited the library at Cambridge University to track down an obscure reference. [...] I was blown away by the fact that a university library could be open 24 hours a day!”

Some might find Weideman’s work – and even his way of working – daunting, but this is definitely not how he experiences it.

“You can work by playing. I’m lucky in the sense that my work is my hobby.”