Innovations in spherical approximation - construction, analysis and applications. The motivating problems for this project come from geophysics, including climate, weather forecasting, planetary gravitation and magnetism, and from coding theory and molecular chemistry. National benefit is expected to arise both from an improved ability to handle problems of key economic importance, and from an enhanced position in the international scientific world, through public presentation in leading journa ....Innovations in spherical approximation - construction, analysis and applications. The motivating problems for this project come from geophysics, including climate, weather forecasting, planetary gravitation and magnetism, and from coding theory and molecular chemistry. National benefit is expected to arise both from an improved ability to handle problems of key economic importance, and from an enhanced position in the international scientific world, through public presentation in leading journals and international conferences, and from direct collaboration with internationally leading scientists from USA, UK and Germany. The project will also increase the pool of trained mathematicians with expertise in areas important for applications.
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The Sakai scheme-Askey table correspondence, analogues of isomonodromy and determinantal point processes. The Australian mathematical sciences enjoys two research groups with active interests on Painleve equations in applied mathematics which are able to address difficult problems. Such a problem is to give a formulation of Sakai's 2001 classification of the Painleve equations in a form most suitable for applications. For this links will be made with a seemingly distinct area of mathematics - t ....The Sakai scheme-Askey table correspondence, analogues of isomonodromy and determinantal point processes. The Australian mathematical sciences enjoys two research groups with active interests on Painleve equations in applied mathematics which are able to address difficult problems. Such a problem is to give a formulation of Sakai's 2001 classification of the Painleve equations in a form most suitable for applications. For this links will be made with a seemingly distinct area of mathematics - the Askey table from the theory of hypergeometric orthogonal polynomials. A number of tractable PhD projects are suggested by this proposal.Read moreRead less
New Directions in Non-linear Mathematical Asymptotics. Major challenges such as predicting epidemics or modelling cancer rely on our understanding of simple mathematical models with extremely complicated solutions. The first and only model in the literature to reproduce the three-phase cycle of immune response in HIV/AIDS was based on cellular automata. Its results are extremely sensitive to infinitesimally small changes in parameters. Yet, no technique exists to study such variation in cellular ....New Directions in Non-linear Mathematical Asymptotics. Major challenges such as predicting epidemics or modelling cancer rely on our understanding of simple mathematical models with extremely complicated solutions. The first and only model in the literature to reproduce the three-phase cycle of immune response in HIV/AIDS was based on cellular automata. Its results are extremely sensitive to infinitesimally small changes in parameters. Yet, no technique exists to study such variation in cellular automata. This research will provide new methods for prediction and analysis of such models. Read moreRead less
Exploratory Experimentation and Computation in the Mathematical Sciences: Theory and Practice. Seemingly disparate mathematical research projects rely on subtle experimental mathematics methods, and have unveiled weaknesses in current computer algebra systems and symbolic-numeric-graphic tools. The project attacks issues of efficiency, effectiveness, reliability, and certifiability in high-precision mathematical and scientific computation. This will be done by developing enhanced tools for advan ....Exploratory Experimentation and Computation in the Mathematical Sciences: Theory and Practice. Seemingly disparate mathematical research projects rely on subtle experimental mathematics methods, and have unveiled weaknesses in current computer algebra systems and symbolic-numeric-graphic tools. The project attacks issues of efficiency, effectiveness, reliability, and certifiability in high-precision mathematical and scientific computation. This will be done by developing enhanced tools for advanced computation of special functions driven by pursuit of challenging research problems. The focus is on tractable components that arose in prior research on effective high-precision algorithms for multiple integrals, such as arise throughout mathematical physics, number theory and elsewhere.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
Solving inverse problems with Iterative regularisation and convex penalties. This project aims to develop and investigate new computational procedures for the solution of inverse problems which do not have the usual smoothness properties (or source conditions) required for the traditional regularisation methods. Examples of such inverse problems are very common and include image restoration, photo-acoustic tomography and spectroscopy. It is anticipated that this project will substantially extend ....Solving inverse problems with Iterative regularisation and convex penalties. This project aims to develop and investigate new computational procedures for the solution of inverse problems which do not have the usual smoothness properties (or source conditions) required for the traditional regularisation methods. Examples of such inverse problems are very common and include image restoration, photo-acoustic tomography and spectroscopy. It is anticipated that this project will substantially extend the toolbox of methods for such problems utilising ideas from Banach spaces, convex analysis, parallel computing and optimisation. This project is expected to make a substantial contribution to a better understanding of inverse problems and their solution procedures.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE170100171
Funder
Australian Research Council
Funding Amount
$360,000.00
Summary
Towards a mathematical description of magneto-hydrodynamic turbulence. The project aims to better predict magneto-hydrodynamic turbulence than existing empirical models. Turbulence in high-speed flows of electrically conductive fluid sustains magnetic fields in various engineering, geophysical, and astrophysical flows. However, investigations into magneto-hydrodynamic flows have been limited to slow flows, and the application of the results to the actual problems hindered. This project aims to i ....Towards a mathematical description of magneto-hydrodynamic turbulence. The project aims to better predict magneto-hydrodynamic turbulence than existing empirical models. Turbulence in high-speed flows of electrically conductive fluid sustains magnetic fields in various engineering, geophysical, and astrophysical flows. However, investigations into magneto-hydrodynamic flows have been limited to slow flows, and the application of the results to the actual problems hindered. This project aims to improve magneto-hydrodynamic flow control in future energy-generating technology, using theoretical and numerical tools that are mathematically consistent with the high-speed limit of the governing equations. More efficient electric generators could improve Australia’s future energy supply with fewer emissions of global warming gases.Read moreRead less
Novel nonlinear functional analysis methods for singular and impulsive boundary value problems. This project aims to develop innovative functional analysis theories and methods to study various complex nonlinear boundary value problems, with particular focus on singular and impulsive problems. The outcomes of this project aims to enhance Australia’s capability of tackling complex nonlinear science and engineering problems using sophisticated mathematical methods. This project aims to also provid ....Novel nonlinear functional analysis methods for singular and impulsive boundary value problems. This project aims to develop innovative functional analysis theories and methods to study various complex nonlinear boundary value problems, with particular focus on singular and impulsive problems. The outcomes of this project aims to enhance Australia’s capability of tackling complex nonlinear science and engineering problems using sophisticated mathematical methods. This project aims to also provide engineers and scientists with a theoretical base and simulation technique for the study and optimal control of impulsive systems and processes involving nonlinear singularity.Read moreRead less
Theory and applications of three dimensional fractal transformations. The purpose of this project is to develop the theory and algorithms for a new class of continuous mappings between fractals. Outcomes include a better understanding of fractals, substantially better algorithms for fractal compression and many new applications.
Fundamental Studies in System Identification. To operate a dynamic system such as a chemical process plant or an economy one needs two things; the equations describing the system; a way of regulating the system to provide desired outcomes. System identification provides the first; control engineering design provides the second. This proposal addresses three important problems in system identification and control. Firstly since the equations can never be known precisely we aim to determine what i ....Fundamental Studies in System Identification. To operate a dynamic system such as a chemical process plant or an economy one needs two things; the equations describing the system; a way of regulating the system to provide desired outcomes. System identification provides the first; control engineering design provides the second. This proposal addresses three important problems in system identification and control. Firstly since the equations can never be known precisely we aim to determine what is the best one can do? Secondly to provide then tight error bounds for the control design;
thirdly to develop new methods for some hitherto unresolved problems in system identification.Read moreRead less