A new asymptotic toolbox for nonlinear discrete systems and particle chains. This project aims to pioneer a mathematical toolbox of new asymptotic techniques for discrete systems driven by vanishingly small influences. The purpose of these techniques is to permit the asymptotic study of discrete problems in which significant effects originate due to subtle causes that are invisible to existing asymptotic methods. Discrete systems play a significant role in modern applied mathematics, and it is v ....A new asymptotic toolbox for nonlinear discrete systems and particle chains. This project aims to pioneer a mathematical toolbox of new asymptotic techniques for discrete systems driven by vanishingly small influences. The purpose of these techniques is to permit the asymptotic study of discrete problems in which significant effects originate due to subtle causes that are invisible to existing asymptotic methods. Discrete systems play a significant role in modern applied mathematics, and it is vital that mathematical tools be designed in order to explore their behaviour. The aim of this project is to open new pathways for resolving open scientific problems, providing benefits such as understanding the energy dissipation of particle chains and granular lattices contained in small-scale technological components.Read moreRead less
High Dimensional Computation and Uncertainty. This project aims to establish powerful computational methods for high-dimensional problems - methods that are rigorous, and carefully tailored to specific applications, from physics, environment, manufacturing and finance, and often driven by uncertainty. The project will generate new knowledge in the area of high-dimensional computation, and develop technological innovations in key areas of science and industry. Expected outcomes include improved c ....High Dimensional Computation and Uncertainty. This project aims to establish powerful computational methods for high-dimensional problems - methods that are rigorous, and carefully tailored to specific applications, from physics, environment, manufacturing and finance, and often driven by uncertainty. The project will generate new knowledge in the area of high-dimensional computation, and develop technological innovations in key areas of science and industry. Expected outcomes include improved control of uncertainty in industry, enhanced international and interdisciplinary collaborations, and significant publications and presentations in international forums. The technological advancements will help boost Australia's position as a world leader in innovation.Read moreRead less
Expanding and linking random matrix theory. Fundamental to random matrix theory are certain universality laws, holding in scaling limits to infinite matrix size. A basic question is to quantify the rate of convergence to the universal laws. The analysis of data for the Riemann zeros from prime number theory, and of the spectral form factor probe of chaos in black hole physics, are immediate applications. An analysis involving integrable structures holding for finite matrix size and their asympt ....Expanding and linking random matrix theory. Fundamental to random matrix theory are certain universality laws, holding in scaling limits to infinite matrix size. A basic question is to quantify the rate of convergence to the universal laws. The analysis of data for the Riemann zeros from prime number theory, and of the spectral form factor probe of chaos in black hole physics, are immediate applications. An analysis involving integrable structures holding for finite matrix size and their asymptotics is proposed, allowing the rate to be quantified for a large class of model
ensembles, and providing predictions in the various applied settings. The broad project is to be networked with researchers in the Asia-Oceania region, with the aim of establishing leadership status for Australia.Read moreRead less
An optimisation-based framework for non-classical Chebyshev approximation. This project aims to solve open mathematical problems in multivariate and piecewise polynomial approximations, two directions that correspond to fundamental obstacles to extending classical approximation results. Through an innovative combination of optimisation and algebraic technique, the project intends to develop foundations for new results in approximation theory, and new insights into other areas of mathematics, mos ....An optimisation-based framework for non-classical Chebyshev approximation. This project aims to solve open mathematical problems in multivariate and piecewise polynomial approximations, two directions that correspond to fundamental obstacles to extending classical approximation results. Through an innovative combination of optimisation and algebraic technique, the project intends to develop foundations for new results in approximation theory, and new insights into other areas of mathematics, most notably optimisation. The techniques and methods developed should also have significant benefits in the many disciplines where approximation problems appear, such as engineering, physics or data mining. The research outputs resulting from this project will be used in a wide range of fields to help implement programs, policies and improve decision making.Read moreRead less
Regularisation methods of inverse problems: theory and computation. This project aims to investigate regularisation methods for inverse problems which are ill-posed in the sense that their solutions depend discontinuously on the data. When only noisy data is available, regularisation methods define stable approximate solutions by replacing the original inverse problem with a family of well-posed neighbouring problems monitored by a so-called regularisation parameter. The project expects to devel ....Regularisation methods of inverse problems: theory and computation. This project aims to investigate regularisation methods for inverse problems which are ill-posed in the sense that their solutions depend discontinuously on the data. When only noisy data is available, regularisation methods define stable approximate solutions by replacing the original inverse problem with a family of well-posed neighbouring problems monitored by a so-called regularisation parameter. The project expects to develop purely data-driven rules to choose the regularisation parameter and show how they work in theory, and in practice. It will also develop convex framework, acceleration strategies as well as preconditioning and splitting ideas to design efficient regularisation solvers.Read moreRead less
Geometric analysis of nonlinear systems. Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elus ....Geometric analysis of nonlinear systems. Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elusive solutions of discrete and higher-dimensional nonlinear systems. This should provide significant benefits, such as new mathematical knowledge, innovative techniques, enhanced scientific capacity in Australia.Read moreRead less