CARMA Research Group
Mathematical Analysis and Systems Theory
Leader: Jeff Hogan
The group has wide-ranging expertise in many branches of mathematical analysis, including harmonic, functional, nonlinear, nonsmooth and geometric analysis. In particular, our members work in:
* Fourier analysis -- wavelets and other tools for signal and image processing;
* Clifford analysis -- higher-dimensional function theory with applications to multichannel mutivariable signal processing
* Geometry in Banach spaces and convex metric spaces
* Numerical methods for partial differential equations, including finite element methods
* Approximation theory
* Variational methods, with applications to image processing
* Control theory, with applications to energy systems, climate economics and cyber-security
* Geometric analysis -- calculus of variations
* Optimisation (theoretical and numerical) with applications to signal and image processing, including compresssed sensing
* Experimental mathematics, with emphasis on optimisation
Systems theory is, in a broad sense, the theory of mathematical processes that evolve with time. These processes, also called dynamical systems, are often described by differential or difference equations. In these cases the systems are referred to as continuous and, respectively, discrete-time dynamical systems. If a system can be influenced from the outside, it is called a control system. The expertise and interests of our research group cover stability theory of differential and difference inclusions, which are generalisations of the aforementioned concepts. This goes along with applications in areas such as
* climate economics
Our experience spans small and fast one-chip implementations to heterogeneous large-scale systems.
VG10, Mathematics Building
"The Role of Stokes lines in Physical Systems"
Dr Chris Lustri
Systems with small parameters are often studied using asymptotic techniques. Despite the ubiquity of these techniques, many classical asymptotic methods are unable to capture behaviour that occurs on an exponentially small scale, which lies "beyond all orders" of power series in the small parameter. Typically this does not cause any issues; this behaviour is too small to have a measurable impact on the overall behaviour of the system. I will showcase two systems in which exponentially small contributions have a significant effect on the overall system behaviour.
The first system, which I will discuss in detail, will be nonlinear waves propagating through particle chains with periodic masses. I will show that it is typically possible for Toda and FPUT lattices for certain combinations of parameters - determined by the exponentially small system behaviour - to produce solitary waves that propagate indefinitely. The second system, which I will discuss more briefly, will be the shape of bubbles in a steadily translating Hele-Shaw cell. By studying exponentially small effects, it is possible to construct exotic bubble shapes which correspond to recent laboratory experiments.
SR202, SR Building
"Optimization Methods for Inverse Problems from Imaging"
Prof Ke Chen
Optimization is often viewed as an active and yet mature research field. However the recent and rapid development in the emerging field of Imaging Sciences has provided a very rich source of new problems as well as big challenges for optimization. Such problems having typically non-smooth and non-convex functionals demand urgent and major improvements on traditional solution methods suitable for convex and differentiable functionals.
This talk presents a limited review of a set of Imaging Models which are investigated by the Liverpool group as well as other groups, out of the huge literature of related works. We start with image restoration models regularised by the total variation and high order regularizers. We then show some results from image registration to align a pair of images which may be in single-modality or multimodality with the latter very much non-trivial. Next we review the variational models for image segmentation. Finally we show some recent attempts to extend our image registration models from more traditional optimization to the Deep Learning framework.
Joint work with recent and current collaborators including D P Zhang, A Theljani, M Roberts, J P Zhang, A Jumaat, T Thompson.