Multiscale Singularly Perturbed Control Systems. We propose to develop a unified averaging technique to analyse deterministic and stochastic multiscale singularly perturbed control systems. Such systems arise as mathematical models of real-world dynamical systems in which state variables can change their values with the rates of different orders of magnitude. The technique is based on the assumption that the system, which would describe the dynamics of the fast state variables if slow ones were ....Multiscale Singularly Perturbed Control Systems. We propose to develop a unified averaging technique to analyse deterministic and stochastic multiscale singularly perturbed control systems. Such systems arise as mathematical models of real-world dynamical systems in which state variables can change their values with the rates of different orders of magnitude. The technique is based on the assumption that the system, which would describe the dynamics of the fast state variables if slow ones were frozen, possesses certain ergodicity properties expressed in the existence of its limit occupational measures set. Conditions for the existence of such a set will be studied and its structure will be described.Read moreRead less
Occupational Measures Approach to Long Run Average and Singularly Perturbed Optimal Control Problems. Problems of optimal control of long-run average and singularly perturbed systems arise in many applications. The project will lead to the development of new linear programming based techniques for analyzing these problems (including problems intractable so far) and finding their numerical solutions. The new techniques will have a potential to be further developed into software that can benefit A ....Occupational Measures Approach to Long Run Average and Singularly Perturbed Optimal Control Problems. Problems of optimal control of long-run average and singularly perturbed systems arise in many applications. The project will lead to the development of new linear programming based techniques for analyzing these problems (including problems intractable so far) and finding their numerical solutions. The new techniques will have a potential to be further developed into software that can benefit Australian industries and technologies. The proposed topic is in the focus of interest of many eminent researchers around the world and the dissemination of our results will further improve Australia's standing in the international research community. Read moreRead less
Effective and accurate model dynamics, deterministic and stochastic, across multiple space and time scales. A persistent feature of complex systems in engineering and science is the emergence of macroscopic, coarse grained, coherent behaviour from the interactions of microscopic agents (molecules, cells, grains) and with their environment. In current modeling, ranging from ecology to materials science, the underlying microscopic mechanisms are often known, but the closures to translate microscal ....Effective and accurate model dynamics, deterministic and stochastic, across multiple space and time scales. A persistent feature of complex systems in engineering and science is the emergence of macroscopic, coarse grained, coherent behaviour from the interactions of microscopic agents (molecules, cells, grains) and with their environment. In current modeling, ranging from ecology to materials science, the underlying microscopic mechanisms are often known, but the closures to translate microscale knowledge to a system level macroscopic description are rarely available in closed form. Our novel methodology will explore this stumbling block, and promises to radically change the modeling, exploration and understanding of multiscale complex system behaviour.Read moreRead less
Modelling of multiscale systems in engineering and science supports large-scale equation-free simulations and analysis. A persistent feature of complex systems in engineering and science is the emergence of macroscopic, coarse grained, coherent behaviour from the interactions of microscopic agents (molecules, cells) and with their environment. In current modeling, ranging from ecology to materials science, the underlying microscopic mechanisms are known, but the closures to translate microscale ....Modelling of multiscale systems in engineering and science supports large-scale equation-free simulations and analysis. A persistent feature of complex systems in engineering and science is the emergence of macroscopic, coarse grained, coherent behaviour from the interactions of microscopic agents (molecules, cells) and with their environment. In current modeling, ranging from ecology to materials science, the underlying microscopic mechanisms are known, but the closures to translate microscale knowledge to a system level macroscopic description are rarely available in closed form. Our novel, equation free, computational methodologies will circumvent this stumbling block, and promises to radically change the modeling, exploration and understanding of complex system behavior. We continue to develop this powerful computational methodology. Read moreRead less
Risk Measures and Management in Finance and Actuarial Science Under Regime-Switching Models. New models for assessing and managing risk of financial products will place Australia at the forefront of risk management. The work will also sustain the competitive edge of Australia as one of the major financial centres in the Asia-Pacific region through enhancing both the theory and practice of financial risk management. The project outcome will also benefit to the country in other areas of risk, for ....Risk Measures and Management in Finance and Actuarial Science Under Regime-Switching Models. New models for assessing and managing risk of financial products will place Australia at the forefront of risk management. The work will also sustain the competitive edge of Australia as one of the major financial centres in the Asia-Pacific region through enhancing both the theory and practice of financial risk management. The project outcome will also benefit to the country in other areas of risk, for example, environment risk, climate change, and energy and security problems.Read moreRead less
Dynamic risk measures. Exposure to risk is a pervasive problem. The results will be of importance for financial institutions when they estimate their exposure to risk. Other applications will be to determine the level of risk from a terrorist attack or regional instability. Companies wish to allocate resources to minimize their exposure to adverse events.
The fundamental equations for inversion of operator pencils. This project seeks to deepen understanding of how complex systems may be significantly changed by incremental changes to ambient conditions. Mathematical models of complex systems (climate change processes, optimal driving strategies, efficient distribution policies, effective search routines) often depend on key parameters. If small perturbations to the parameters cause large changes to the solution, then the perturbations are said to ....The fundamental equations for inversion of operator pencils. This project seeks to deepen understanding of how complex systems may be significantly changed by incremental changes to ambient conditions. Mathematical models of complex systems (climate change processes, optimal driving strategies, efficient distribution policies, effective search routines) often depend on key parameters. If small perturbations to the parameters cause large changes to the solution, then the perturbations are said to be singular. This project aims to reveal the underlying mathematical structures and develop new computational algorithms to analyse a general class of perturbed systems both locally near an isolated singularity and globally. It plans to use these algorithms to solve systems of equations, calculate generalised inverse operators, examine perturbed Markov processes, and estimate exit times from meta-stable states in stochastic population dynamics.Read moreRead less
Operator Integrals and Derivatives. The project is a contribution to the study of non-commutative differential and integral calculus. The novelty of the present project lies in the study of smoothness properties of functions whose domains and ranges are spaces of unbounded, non-commuting operators on some Hilbert space. Our general approach will be based on a detailed investigation of properties of double operator integrals, which permit smoothness estimates of operator-functions. It can be expe ....Operator Integrals and Derivatives. The project is a contribution to the study of non-commutative differential and integral calculus. The novelty of the present project lies in the study of smoothness properties of functions whose domains and ranges are spaces of unbounded, non-commuting operators on some Hilbert space. Our general approach will be based on a detailed investigation of properties of double operator integrals, which permit smoothness estimates of operator-functions. It can be expected that the new techniques generated will find further application in areas of mathematical physics and non-commutative geometry related to quantized calculus.
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Non-commutative analysis and differential calculus. This project is in an area of central mathematical importance and will lead to important scientific advances that will keep Australia at the forefront internationally in this field of research. There is an emphasis on international networking and we will collaborate with leading researchers in USA and France.
Structured barrier and penalty functions in infinite dimensional optimisation and analysis. Very large scale tightly-constrained optimisation problems are ubiquitous and include water management, traffic flow, and imaging at telescopes and hospitals. Massively parallel computers can solve such problems and provide physically realisable solution only if subtle design issues are mastered. Resolving such issues is the goal of this project.