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
Discovery Early Career Researcher Award - Grant ID: DE170100222
Funder
Australian Research Council
Funding Amount
$313,964.00
Summary
Optimal adaptivity for uncertainty quantification. This project aims to use an adaptive mesh refinement algorithm to improve the ratio of approximation accuracy versus computational time. Partial differential equations with random coefficients are crucial in simulating groundwater flow, structural stability and composite materials, but their numerical approximation is difficult and time consuming. Advances in adaptive mesh refinement theory allow full analysis and mathematical understanding of t ....Optimal adaptivity for uncertainty quantification. This project aims to use an adaptive mesh refinement algorithm to improve the ratio of approximation accuracy versus computational time. Partial differential equations with random coefficients are crucial in simulating groundwater flow, structural stability and composite materials, but their numerical approximation is difficult and time consuming. Advances in adaptive mesh refinement theory allow full analysis and mathematical understanding of the convergence behaviour of the proposed algorithm. The project intends to develop a theory of adaptive algorithms and freely available software for their reliable (and mathematically underpinned) simulation which could solve problems beyond the capabilities of even the most powerful computers.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
Australian Laureate Fellowships - Grant ID: FL120100094
Funder
Australian Research Council
Funding Amount
$3,184,657.00
Summary
Geometric construction of critical solutions of nonlinear systems. Whether we are looking at waves on a beach, the dispersal of herds of animals in a landscape, or the interaction of black holes, their patterns of movement rely on rules expressed by non-linear mathematical models. This project will aim to create new mathematical methods to describe the solutions of non-linear systems, which are ubiquitous in modern science.
Critical solutions of nonlinear systems. Whether we are looking at waves on a beach, the dispersal of herds of animals in a landscape, or the interaction of black holes, their patterns of movement rely on rules expressed by nonlinear mathematical models. This project aims to create new mathematical methods to describe critical solutions of nonlinear systems, which are ubiquitous in modern science.
Reflection Groups and Discrete Dynamical Systems. This project aims to solve long-standing problems in discrete dynamical systems that are of particular interest to physics, by using reflection groups to reveal unexpected geometric insights. Mathematics has the power to abstract crucial patterns from complex observations. Symmetries familiar in the real world, like the hexagonal patterns of honeycombs, arise inside convoluted structures in high-dimensional systems. By revealing the structure of ....Reflection Groups and Discrete Dynamical Systems. This project aims to solve long-standing problems in discrete dynamical systems that are of particular interest to physics, by using reflection groups to reveal unexpected geometric insights. Mathematics has the power to abstract crucial patterns from complex observations. Symmetries familiar in the real world, like the hexagonal patterns of honeycombs, arise inside convoluted structures in high-dimensional systems. By revealing the structure of space-filling polytopes in integrable systems, the project seeks to find sought-after reductions of high-dimensional discrete models to two dimensions. Expected outputs include new reductions to discrete Painlevé equations, new circle patterns useful for computer graphics and discrete holomorphic functions.Read moreRead less
Quantifying uncertainty: innovations in high dimensional computation. High dimensional problems (problems in which there are hundreds or thousands of continuous variables) arise in many applications, from ground water flow to mathematical physics and finance. They typically present major challenges to computational resources and serious mathematical challenges in devising new and improved methods and in proving that they are effective. The aim of this project is to develop new computational meth ....Quantifying uncertainty: innovations in high dimensional computation. High dimensional problems (problems in which there are hundreds or thousands of continuous variables) arise in many applications, from ground water flow to mathematical physics and finance. They typically present major challenges to computational resources and serious mathematical challenges in devising new and improved methods and in proving that they are effective. The aim of this project is to develop new computational methods and theory for high dimensional problems, and to apply these methods to significant applications. The results are expected to allow faster and more accurate solution of problems of growing importance.Read moreRead less
Novel technology for enhanced coal seam gas production utilising mechanisms of stimulated cleat permeability through graded particle injection. This cross-disciplinary project will develop a new integrated technology for well productivity enhancement in coal seam gas, shale, tight gas and geothermal reservoirs - the world’s fastest growing unconventional clean energy resources. It will improve our understanding of the multi scale physics of natural gas and energy production.
Towards a science of high dimensional computation. This project aims to establish scientifically precise methods for high dimensional problems - methods that are mathematically rigorous, empirically tested, and carefully tailored to specific modern applications across physics, environment, and finance. This project expects to generate new knowledge in the area of high dimensional computation and to develop technological innovations in key areas of science and industry. Expected outcomes of this ....Towards a science of high dimensional computation. This project aims to establish scientifically precise methods for high dimensional problems - methods that are mathematically rigorous, empirically tested, and carefully tailored to specific modern applications across physics, environment, and finance. This project expects to generate new knowledge in the area of high dimensional computation and to develop technological innovations in key areas of science and industry. Expected outcomes of this project include enhanced international and interdisciplinary collaborations, and significant publications and presentations in international forums. These technological advancements will help boost Australia’s position as a world leader in creativity and innovation.Read moreRead less