Using computational techniques to approximate solutions to physical models is common, but these techniques are not worth much if not backed up by serious mathematical analysis – it is not rare to see algorithms that are ill-designed or ill-used, and thus lead to unrealistic solutions. Modern opportunities test the boundaries of old algorithms, and ask for new numerical or symbolic techniques and analysis to be performed. This must be founded on solid mathematical theories if we want to go beyond a trial-and-error process.
The AustMS SIG we propose revolves around the design and analysis, using mathematical rigour, of numerical algorithms for models based on differential equations and mathematical optimisation.
At the 2006 annual meeting, a special session of the AustMS annual meeting was held around computational mathematics and optimisation. Since then, special sessions involving these areas have been regularly held at the AustMS annual meetings. In the last few years, a regular Computational Maths session has been organised by B. P. Lamichhane and Q. T. Le Gia, with 20+ participants each year on average. Similarly, since 2009 the organisers of the ANZIAM SigmaOpt group have run successful Optimisation/Control sessions at the annual AustMS meeting, with 15-30 participants each year; in 2015 the session had to refuse talks due to the lack of available slots. Several other sessions at AustMS meetings also indicate an interest in mathematical topics strongly related to optimisation and computation: industrial mathematics, DSTO session, mathematical biology, mathematics of medical imaging, natural resource mathematics, variational analysis and optimisation, to name a few from the 2015 meeting.
All this demonstrates the existence of a critical mass of AustMS members interested in the mathematical analysis of numerical and optimisation techniques.
The AustMS SIG purpose will be to organise events on these topics, within and outside the AustMS annual meetings. There is a strong interface between this SIG and the ANZIAM groups CMG and SigmaOpt, and we want to make sure that this interface exists in the day-to-day working of all these groups. Both CMG and SigmaOpt, however, have members whose fields of interest are largely disconnected from the AustMS. Moreover, some research fields in numerical analysis and optimisation are strongly grounded in theoretical mathematics, and thus naturally fit within the AustMS. There is, therefore, room for a separate group within the AustMS. This group will help make mathematics for computation more visible in the mathematical community, and that will foster the development of more comprehensive, systematic and rigorous mathematical approaches to numerical algorithms and techniques used in interdisciplinary applications.