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
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
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
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
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
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
Very high dimensional computation - the new frontier in numerical analysis. High-dimensional problems, involving hundreds or thousands of variables, arise in applications from finance, health statistics and oil reservoir modelling to physics and chemistry. This project aims to develop the science of high-dimensional computation, as driven by important applications such as the flow of groundwater through a porous material.