Discovery Early Career Researcher Award - Grant ID: DE120101707
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Regularisation methods for solving nonlinear ill-posed inverse problems. Nonlinear inverse problems arise in numerous applications and their stable resolutions require regularisation methods. This project will develop various efficient solvers by using optimisation tools and Newton type procedures and consider their convergence properties. The methods will be applied to practical problems including the tomography techniques.
Existence and Stability of a Model for Three-Dimensional Toroidal Plasma Equilibria. There is great physical interest in modelling strongly non-axisymmetric toroidal plasmas, but fundamental existence problems have made rigorous numerical analysis so far impossible. We seek to overcome this by investigating a class of idealized, but physically motivated, magnetohydrodynamic equilibria with stepped pressure profiles for which existence in the neighbourhood of axisymmetry has been proven. We will ....Existence and Stability of a Model for Three-Dimensional Toroidal Plasma Equilibria. There is great physical interest in modelling strongly non-axisymmetric toroidal plasmas, but fundamental existence problems have made rigorous numerical analysis so far impossible. We seek to overcome this by investigating a class of idealized, but physically motivated, magnetohydrodynamic equilibria with stepped pressure profiles for which existence in the neighbourhood of axisymmetry has been proven. We will (i) develop numerical techniques to extend these piece-wise Beltrami states far away from axisymmetry (ii) develop practical tests to determine when existence breaks down (iii) analyze the frequency spectrum of small oscillations about such equilibria (iv) extend the model to two-fluid MHD.Read moreRead less
Sparse grid approximations and fitting using generalised combination techniques. Sparse grid techniques provide an effective tool to deal with the
computational curse of dimensionality which is a constant challenge in
modelling complex data. The proposed research is aimed at the
development and analysis of algorithms for data fitting with sparse
grids using variants of the combination technique. The outcome of the
research is a theory which will provide insights in the applicability,
limit ....Sparse grid approximations and fitting using generalised combination techniques. Sparse grid techniques provide an effective tool to deal with the
computational curse of dimensionality which is a constant challenge in
modelling complex data. The proposed research is aimed at the
development and analysis of algorithms for data fitting with sparse
grids using variants of the combination technique. The outcome of the
research is a theory which will provide insights in the applicability,
limitations and the convergence properties of the proposed
algorithms. The outcomes will be widely applicable in modelling of
large scale and complex data as is encountered in areas of
bioinformatics, physics and experimental studies of complex systems.
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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
David and Goliath - what planets can do to the stars that created them. We used to think that when stars expand during their old age, they would destroy all their close-by planets. Today we know that if a star swallows a Jupiter-like planet it can suffer indigestion. The project will study how star-planet interactions takes place, determine their impact on the lives of stars and glimpse at the future of our own solar system.
Robust numerical solution of partial differential equations on petascale computer systems with applications to tsunami modelling and plasma physics. The project will apply new mathematical ideas to exploit the unprecedented computational resources provided by the next generation of high performance computers. The resulting techniques and software will form a key component for the science needed to understand the workings of complex dynamical systems, such as tsunamis and fusion reactors.
Harnessing Spherical Geometry in Scientific and Engineering Data Processing. Spherical information underpins many natural phenomena, ranging from the distribution of galaxies in the Universe to the connectivity and neuronal activation in the human brain. Current major investments in scientific and medical instrumentation do not efficiently collect and process the massive amounts of data because they do not properly utilise its inherent spherical geometry. Through harnessing spherical geometry, t ....Harnessing Spherical Geometry in Scientific and Engineering Data Processing. Spherical information underpins many natural phenomena, ranging from the distribution of galaxies in the Universe to the connectivity and neuronal activation in the human brain. Current major investments in scientific and medical instrumentation do not efficiently collect and process the massive amounts of data because they do not properly utilise its inherent spherical geometry. Through harnessing spherical geometry, this project aims to address the above shortcomings and to provide advances across all these application domains. By collecting and processing data more efficiently, with greater fidelity, and by revealing features currently hidden, the methods developed are expected to see the full benefit from the instrumentation capturing this data.Read moreRead less
Global Economic Consequences of Korean Re-unification. This research aims to model the economic implications of Korean re-unification and explore the spill-overs to key economies in the region, including Australia. The approach is understood to be first to develop a model for the existing North Korean economy in collaboration with South Korean researchers at Korea University. This model is intended to be embedded in an existing global economic model. The project aims to then explore scenarios of ....Global Economic Consequences of Korean Re-unification. This research aims to model the economic implications of Korean re-unification and explore the spill-overs to key economies in the region, including Australia. The approach is understood to be first to develop a model for the existing North Korean economy in collaboration with South Korean researchers at Korea University. This model is intended to be embedded in an existing global economic model. The project aims to then explore scenarios of integration between North and South Korea focusing on changing production structure, development in human capital formation, technology transfer, immigration flows within a unified Korea and the impact on trade and financial flows that may result from different scenarios of how unification will proceed.Read moreRead less
Numerical Algorithms for Solving Convex Optimization Problems Arising in Systems and Control Theory. The need to optimize occurs frequently in engineering applications. Typically one has a set of constraints specifying what solutions are allowable or meet design specifications and one would like to choose from these allowable solutions one which is optimal with respect to some meaningful metric. Such optimization problems tend to be rather complicated and must be solved numerically. This project ....Numerical Algorithms for Solving Convex Optimization Problems Arising in Systems and Control Theory. The need to optimize occurs frequently in engineering applications. Typically one has a set of constraints specifying what solutions are allowable or meet design specifications and one would like to choose from these allowable solutions one which is optimal with respect to some meaningful metric. Such optimization problems tend to be rather complicated and must be solved numerically. This project is concerned with creating improved numerical algorithms for solving particular important classes of optimization problems that arise in systems and control theory.Read moreRead less
Regularisation methods of inverse problems: theory and computation. This project aims to investigate regularisation methods for inverse problems which are ill-posed in the sense that their solutions depend discontinuously on the data. When only noisy data is available, regularisation methods define stable approximate solutions by replacing the original inverse problem with a family of well-posed neighbouring problems monitored by a so-called regularisation parameter. The project expects to devel ....Regularisation methods of inverse problems: theory and computation. This project aims to investigate regularisation methods for inverse problems which are ill-posed in the sense that their solutions depend discontinuously on the data. When only noisy data is available, regularisation methods define stable approximate solutions by replacing the original inverse problem with a family of well-posed neighbouring problems monitored by a so-called regularisation parameter. The project expects to develop purely data-driven rules to choose the regularisation parameter and show how they work in theory, and in practice. It will also develop convex framework, acceleration strategies as well as preconditioning and splitting ideas to design efficient regularisation solvers.Read moreRead less