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
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
Creating Hybrid Exponential Asymptotics for use with Computational Data. Asymptotic analysis is a vital tool for studying small influences with critical effects. This project aims to create an innovative fully-automated asymptotic framework for studying phenomena which are invisible to classical approximation methods, using new ideas from asymptotics and numerical complex analysis. The outcome will be the first framework that can be used on data from numerical simulations or real-life measuremen ....Creating Hybrid Exponential Asymptotics for use with Computational Data. Asymptotic analysis is a vital tool for studying small influences with critical effects. This project aims to create an innovative fully-automated asymptotic framework for studying phenomena which are invisible to classical approximation methods, using new ideas from asymptotics and numerical complex analysis. The outcome will be the first framework that can be used on data from numerical simulations or real-life measurements, and which can be applied automatically without hands-on expert input. It will be used to design submerged structures and efficient vessels with minimal energy loss from surface waves. Expected benefits include making powerful methods accessible to scientists, and new paths for energy-efficient industrial design.Read moreRead less
High Dimensional Approximation, Learning, and Uncertainty. This project aims to develop next-generation computational methods for complex problems in science and engineering that have many uncertain parameters, using advanced high-dimensional strategies and deep learning to enhance computational speed. The significance of the project is that these methods will help address important applications that at present are not feasible or at the edge of feasibility. The expected outcomes are powerful me ....High Dimensional Approximation, Learning, and Uncertainty. This project aims to develop next-generation computational methods for complex problems in science and engineering that have many uncertain parameters, using advanced high-dimensional strategies and deep learning to enhance computational speed. The significance of the project is that these methods will help address important applications that at present are not feasible or at the edge of feasibility. The expected outcomes are powerful methods that will be mathematically rigorous and suitable for a wide variety of applications. The benefits are that the project will boost Australia’s position as a leader in innovation, and contribute to future developments over a wide area, from aerospace engineering to personalised computational oncology.Read moreRead less
Approximation theory of structured neural networks . Mathematical theory for deep learning has been desired due to the power applications of deep neural networks to deal with big data in various practical domains. The main difficulty lies in the structures and architectures imposed to networks designed for specific learning tasks. Neither the classical approximation theory nor the recent one for depths of ReLU neural networks can be applied due to the structures imposed for processing large dime ....Approximation theory of structured neural networks . Mathematical theory for deep learning has been desired due to the power applications of deep neural networks to deal with big data in various practical domains. The main difficulty lies in the structures and architectures imposed to networks designed for specific learning tasks. Neither the classical approximation theory nor the recent one for depths of ReLU neural networks can be applied due to the structures imposed for processing large dimensional data such as natural images of tens of thousands of dimensions. This project aims at an approximation theory for structured neural networks. We plan to establish mathematical theories for deconvolution with deep convolutional neural networks, operator learning, and spectral graph networks. Read moreRead less
Mathematics in the round - the challenge of computational analysis on spheres. Real world problems formulated on spheres (including physical problems for the whole earth) provide many difficult challenges. This project aims at developing algorithms to solve problems on spheres in two and higher dimensions, with applications ranging from geophysics to signal analysis.
Liberating the Dimension - Frontier Technologies for Very High Dimensional Problems. Real world problems tend to involve an enormous number of variables (the dimension) - hundreds, thousands, millions, or even infinite. Very high dimensional problems arise in finance, statistics, physics and many other application areas. For example, in option pricing, maximum likelihood estimation in health statistics, and porous flow through a random medium. In this project, the theory of very high dimensional ....Liberating the Dimension - Frontier Technologies for Very High Dimensional Problems. Real world problems tend to involve an enormous number of variables (the dimension) - hundreds, thousands, millions, or even infinite. Very high dimensional problems arise in finance, statistics, physics and many other application areas. For example, in option pricing, maximum likelihood estimation in health statistics, and porous flow through a random medium. In this project, the theory of very high dimensional computation will be developed in step with applications, with the applications being chosen to help in shaping the theory. The project involves interdisciplinary collaborations and aims to develop frontier technologies for tackling realistic high dimensional problems.Read moreRead less
Asymptotics of the exponentially small. Asymptotic analysis plays a vital role in studying the complex interfacial dynamics that are fundamental for practical problems in fluid mechanics such as the withdrawal of oil and gas from underground reservoirs and the optimal design of ship hulls to minimise wave drag. These applications exhibit extremely small physical effects that may be crucially important but cannot be described using classical asymptotic analysis. This project will develop state of ....Asymptotics of the exponentially small. Asymptotic analysis plays a vital role in studying the complex interfacial dynamics that are fundamental for practical problems in fluid mechanics such as the withdrawal of oil and gas from underground reservoirs and the optimal design of ship hulls to minimise wave drag. These applications exhibit extremely small physical effects that may be crucially important but cannot be described using classical asymptotic analysis. This project will develop state of the art mathematical techniques in exponential asymptotics to address this deficiency in the classical theory, and provide a deeper understanding of pattern formation, instabilities and wave propagation on the interface between two fluids.Read moreRead less