Annual NSW/ACT ANZIAM Meeting 2015

Lars Grüne, University of Bayreuth, Germany

On conditions under which receding horizon control delivers approximately optimal solutions

Receding horizon control —also known as model predictive control— is a method which solves optimal control problems on infinite or indefinitely long time horizons by iteratively solving problems on relatively short finite time horizons. It can thus be seen as a model reduction technique in time. Clearly, this technique does not apply to every optimal control problem. In this talk we will discuss conditions under which receding horizon control can be shown to yield approximately optimal solutions. A particular emphasis is put on the so-called turnpike property, which we consider as one of the central properties for making receding horizon control work.

Christopher Poulton, University of Technology Sydney, Australia

Interactions between sound and light on the nanoscale

The interaction between electromagnetic and elastodynamic vibrations has a long and distinguished history, dating from the work of Brillouin in the early 20th century. More recently researchers have begun to rediscover these interactions in the context of nanophotonics, in which light is trapped or guided within structures that possess features that are typically as large as the wavelength of light (and sound) in the material. These interactions can lead to several interesting and unusual effects, including “slow-light”, by which the speed of light is reduced to a fraction of its value in vacuum. However at these small scales the mathematics of the different types of waves, and of the forces that cause them to interact, can become complicated, and modelling of the interlinked PDEs is a difficult task. In addition a number of different competing effects, arising from the electronic properties of the light-bearing medium, can have an outsized influence on the interaction. Here we discuss the journey towards a comprehensive and accurate mathematical description of light-sound interactions in nanophotonics, and review recent progress in using these models in on-chip optical waveguides for a range of novel applications.

Ian H Sloan, The University of New South Wales Sydney, Australia

What’s new in high-dimensional integration?
designing integration rules for applications

High-dimensional integration —numerical integration when there are hundreds or thousands of continuous variables— will be an important direction for numerical analysis far into the future. Such problems are arising with increasing frequency, and can be very hard. Much of the focus will be on applications, in mathematical finance, linear models in statistics, and PDE with random coefficients, the latter typified by flow through a random porous medium. What’s new is that for the first time we are beginning to sign rules that are especially suitable for particular applciations.