Advanced computational algorithms for three-dimensional systems. This project deals with the development, analysis and implementation of efficient computer algorithms for a range of complex three dimensional systems. Major areas of focus are forward and inverse acoustic and electromagnetic scattering; dynamical and evolution processes in water waves and tumour growth; and the solution of mathematical models on spheres (earth). Potential application areas of the project include defence science ....Advanced computational algorithms for three-dimensional systems. This project deals with the development, analysis and implementation of efficient computer algorithms for a range of complex three dimensional systems. Major areas of focus are forward and inverse acoustic and electromagnetic scattering; dynamical and evolution processes in water waves and tumour growth; and the solution of mathematical models on spheres (earth). Potential application areas of the project include defence science; ocean engineering; medical research; meteorology and global environmental sciences.Read moreRead less
Lifting the curse of dimensionality - bringing together the quasi Monte Carlo and sparse grid methods. This project is expected to lead to improved methods for handling high-dimensional problems (i.e. problems with many variables) that arise in finance, statistics, commerce, physics, and many other fields. In turn this could lead to significant economic benefit, especially to high-value service industries such as the finance industry. By strengthening international collaboration, it will also ....Lifting the curse of dimensionality - bringing together the quasi Monte Carlo and sparse grid methods. This project is expected to lead to improved methods for handling high-dimensional problems (i.e. problems with many variables) that arise in finance, statistics, commerce, physics, and many other fields. In turn this could lead to significant economic benefit, especially to high-value service industries such as the finance industry. By strengthening international collaboration, it will also help to maintain Australia's strong position in international research in the mathematical sciences.Read moreRead less
Robust Reformulation Methods. Many decision problems in engineering, business and economics are modeled as nonlinear continuous optimization problems. Often these are made difficult by the existence of constraints. In this project, we reformulate such problems as constrained nonsmooth equations, rather than optimization problems, and develop generalized Newton and quasi-Newton methods for solving them. The expected outcomes of this project include a systematic theory of reformulation methods, ....Robust Reformulation Methods. Many decision problems in engineering, business and economics are modeled as nonlinear continuous optimization problems. Often these are made difficult by the existence of constraints. In this project, we reformulate such problems as constrained nonsmooth equations, rather than optimization problems, and develop generalized Newton and quasi-Newton methods for solving them. The expected outcomes of this project include a systematic theory of reformulation methods, and robust and efficient algorithms for solving some important nonlinear continuous optimization problems. There is high potential for applications in engineering, business and finance.Read moreRead less
Quadratic Support Function Technique to Solving Hard Global Nonconvex Optimization Problems. Optimization techniques are becoming increasingly beneficial to modern Australian society in areas such as manufacturing and commerce by improving technical and management decisions. The proposed research is expected to produce enhanced optimization techniques that can be applied to solve a wider range of important problems too complex to be currently solved. The proposed research also represents an inte ....Quadratic Support Function Technique to Solving Hard Global Nonconvex Optimization Problems. Optimization techniques are becoming increasingly beneficial to modern Australian society in areas such as manufacturing and commerce by improving technical and management decisions. The proposed research is expected to produce enhanced optimization techniques that can be applied to solve a wider range of important problems too complex to be currently solved. The proposed research also represents an international collaboration which will improve Australia's ability to participate effectively in international research and innovation and to produce globally competitive mathematical technologiesRead moreRead less
Continuous Optimization with Linear Matrix Inequality Constraints. The proposed research is expected to lead to new insights and new joint collaborative work for both Autralian and Korean partners. Joining forces of the two teams will ensure that a full range of techniques can be utilized to provide rapid successful research outcomes. The proposed collaboration will give better opportunity to increase the visibility of the work from Korea in Australia, and vice versa. One of the key national be ....Continuous Optimization with Linear Matrix Inequality Constraints. The proposed research is expected to lead to new insights and new joint collaborative work for both Autralian and Korean partners. Joining forces of the two teams will ensure that a full range of techniques can be utilized to provide rapid successful research outcomes. The proposed collaboration will give better opportunity to increase the visibility of the work from Korea in Australia, and vice versa. One of the key national benefits is that the proposed research collaboration will provide extremly fertile ground for training postdoctoral researchers and graduate students in one of the most applicable areas of mathematics.Read moreRead less
Necessary and sufficient conditions for global minimum in multi-extremal global continuous optimization. A basic understanding of the mechanisms for finding local "best" (optimal) solutions has been
achieved through optimization techniques. However, solving global optimization problems, where we may have many local optimal solutions which are not the "absolutely best" (global), is vital for many applications in industry & science, and is intrinsically difficult. The lack of verifiable condition ....Necessary and sufficient conditions for global minimum in multi-extremal global continuous optimization. A basic understanding of the mechanisms for finding local "best" (optimal) solutions has been
achieved through optimization techniques. However, solving global optimization problems, where we may have many local optimal solutions which are not the "absolutely best" (global), is vital for many applications in industry & science, and is intrinsically difficult. The lack of verifiable conditions for a global optimum is a serious limitation. This project will develop verifiable such global optimality conditions for many classes of these problems. A new methodology, functional abstract convexity, developed by CIs and has shown promising results, will be extended and applied for solving these problems.Read moreRead less
A new improved solution to global optimization over multivariate polynomials: Mathematical principles, numerical methods and selected applications. Optimization technology is becoming increasingly beneficial to modern Australian society in areas such as wireless communications and manufacturing by improving performance or reducing costs. Our research will produce enhanced global optimization methodologies, capable of solving a wider range of problems that are currently too complex to be solved. ....A new improved solution to global optimization over multivariate polynomials: Mathematical principles, numerical methods and selected applications. Optimization technology is becoming increasingly beneficial to modern Australian society in areas such as wireless communications and manufacturing by improving performance or reducing costs. Our research will produce enhanced global optimization methodologies, capable of solving a wider range of problems that are currently too complex to be solved. Since global optimization technology is used in many scientific disciplines and modern industrial applications, the research will make many Australian science and industries more competitive. Our research also represents a program of high profile international collaborations that will improve Australia's ability to produce internationally competitive optimization technology.
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Novel Mathematical Technologies for Financial Valuation and Risk. The finance industry is a key industry sector in Australia, with a highly qualified and well-paid work force. The profitability and stability of financial institutions rests in part on their ability to accurately and rapidly value innovative financial products and to quantify risk, yet many contemporary financial products are difficult to value quickly and accurately. This project will bring innovative mathematical technologies ....Novel Mathematical Technologies for Financial Valuation and Risk. The finance industry is a key industry sector in Australia, with a highly qualified and well-paid work force. The profitability and stability of financial institutions rests in part on their ability to accurately and rapidly value innovative financial products and to quantify risk, yet many contemporary financial products are difficult to value quickly and accurately. This project will bring innovative mathematical technologies to bear on complex financial problems. It has the potential to improve both the profitability and stability of Australian financial institutions, and to enhance their role regionally and internationally.Read moreRead less
Constrained and Stable Solutions of Nonlinear and Semismooth Equations. In this project, comprehensive models for designing safe power system parameters will be proposed, efficient algorthms for solving these models will be constructed. The new models and algorithms in this project will provide efficient tools to prevent catastrophic events in power systems, which is related with national security. This project will also strengthen collaboration of Australian applied
mathematians with inter ....Constrained and Stable Solutions of Nonlinear and Semismooth Equations. In this project, comprehensive models for designing safe power system parameters will be proposed, efficient algorthms for solving these models will be constructed. The new models and algorithms in this project will provide efficient tools to prevent catastrophic events in power systems, which is related with national security. This project will also strengthen collaboration of Australian applied
mathematians with international researchers and engineering scientists. This is important for the advance of science and technology in
Australia.Read moreRead less
Approximation, Cubature and Point Designs on Spheres. The sphere is important in fields ranging from geophysics to global climate modelling to chemistry to codes for modern communications. This project aims to strengthen and unify key areas of mathematics on the sphere and at the same time provide methods and constructiions of practical significance. The areas of focus are constructive approximation of functions on the sphere, numerical integration on the sphere, and well distributed sets of poi ....Approximation, Cubature and Point Designs on Spheres. The sphere is important in fields ranging from geophysics to global climate modelling to chemistry to codes for modern communications. This project aims to strengthen and unify key areas of mathematics on the sphere and at the same time provide methods and constructiions of practical significance. The areas of focus are constructive approximation of functions on the sphere, numerical integration on the sphere, and well distributed sets of points on the sphere, including spherical designs.Read moreRead less