Evolutionary Models for Signal Enhancement and Approximation

This thesis deals with nature-inspired evolution processes for the purpose of signal enhancement and approximation. The focus lies on mathematical models which originate from the description of swarm behaviour. We extend existing approaches and show the potential of swarming processes as a modelling tool in image processing. In our work, we discuss the use cases of grey scale quantisation, contrast enhancement, line detection, and coherence enhancement. Furthermore, we propose a new and purely repulsive model of swarming that turns out to describe a specific type of backward diffusion process. It is remarkable that our model provides extensive stability guarantees which even support the utilisation of standard numerics. In experiments, we demonstrate its applicability to global and local contrast enhancement of digital images. In addition, we study the problem of one-dimensional signal approximation with limited resources using an adaptive sampling approach including tonal optimisation. We suggest a direct energy minimisation strategy and validate its efficacy in experiments. Moreover, we show that our approximation model can outperform a method recently proposed by Dar and Bruckstein

Structure-preserving variational schemes for fourth order nonlinear partial differential equations with a Wasserstein gradient flow structure

There is a growing interest in studying nonlinear partial differential equations which constitute gradient flows in the Wasserstein metric and related structure preserving variational discretisations. In this thesis, we focus on the fourth order Derrida-Lebowitz-Speer-Spohn (DLSS) equation, the thin film equation, as well as other fourth order examples. We adapt the minimising movement schemes from implicit Euler (BDF1) to higher order schemes, i.e. backward difference formulae and diagonally implicit Runge-Kutta (DIRK) methods. We prove numerical convergence of discrete solutions of the DIRK2 scheme using a comparison principle type approach with semi-convex based conditions. With basic assumptions including semi-convexity of our energy, verifying that the energy is monotonic in time normally yields convergence of its discrete solution for decreasing time step. However, as in the BDF2 example, for the DIRK2 scheme considered here the energy was not verified to be monotonic (it might be), yet with additional assumptions, convergence is obtained as well as other basic properties of gradient flows. We propose fully discrete schemes which preserve positivity for the DLSS equation, the Thin Film equation and other nonlinear partial differential equations. We present results of numerical experiments confirming improved rates of convergence for higher order schemes. Furthermore, numerical results with non-constant time steps are presented, improving the efficiency of the proposed schemes

Mathematical optimization techniques for demand management in smart grids

The electricity supply industry has been facing significant challenges in terms of meeting the projected demand for energy, environmental issues, security, reliability and integration of renewable energy. Currently, most of the power grids are based on many decades old vertical hierarchical infrastructures where the electric power flows in one direction from the power generators to the consumer side and the grid monitoring information is handled only at the operation side. It is generally believed that a fundamental evolution in electric power generation and supply system is required to make the grids more reliable, secure and efficient. This is generally recognised as the development of smart grids. Demand management is the key to the operational efficiency and reliability of smart grids. Facilitated by the two-way information flow and various optimization mechanisms, operators benefit from real time dynamic load monitoring and control while consumers benefit from optimised use of energy. In this thesis, various mathematical optimization techniques and game theoretic frameworks have been proposed for demand management in order to achieve efficient home energy consumption scheduling and optimal electric vehicle (EV) charging. A consumption scheduling technique is proposed to minimise the peak consumption load. The proposed technique is able to schedule the optimal operation time for appliances according to the power consumption patterns of the individual appliances. A game theoretic consumption optimization framework is proposed to manage the scheduling of appliances of multiple residential consumers in a decentralised manner, with the aim of achieving minimum cost of energy for consumers. The optimization incorporates integration of locally generated and stored renewable energy in order to minimise dependency on conventional energy. In addition to the appliance scheduling, a mean field game theoretic optimization framework is proposed for electric vehicles to manage their charging. In particular, the optimization considers a charging station where a large number of EVs are charged simultaneously during a flexible period of time. The proposed technique provides the EVs an optimal charging strategy in order to minimise the cost of charging. The performances of all these new proposed techniques have been demonstrated using Matlab based simulation studies

Topics in complex multiscale systems: theory and computations of noise-induced transitions and transport in heterogenous media

The present work seeks to address three different problems that have a multiscale nature, we apply different techniques from multiscale analysis to treat these problems. We introduce the field of multiscale analysis and motivate the need for techniques to bridge between scales, presenting the history of some common methods, and an overview of the current state of the field. The remainder of the work deals with the treatment of these problems, one motivated by reaction rate theory, and two from multiphase flow. These superficially have little relation with each other, but the approaches taken share similarities and the results are the same - an average picture of the microscopic description informs the macroscale. In Chapter 2 we address an asymmetric potential with a microscale, showing that the interaction between this microscale and the noise causes a first-order phase transition. This induces a metastable state which we observe and characterise: showing that the stability of this state depends on the strength of the tilt, and that the phase transition is inherently different to the symmetric case. In Chapter 3 we investigate the nucleation and coarsening process of a two-phase flow in a corrugated channel using a Cahn--Hilliard Navier--Stokes model. We show that several flow morphologies can be present depending on the channel geometry and the initial random condition. We rationalise this with a static energy model, predicting the preferential formation of one morphology over another and the existence of a first-order phase-transition from smooth slug flow to discontinuous motion when the channel is strongly corrugated. In Chapter 4 we address a model for interfacial flows in porous geometries, formulating an finite-element model for the equations. Within this framework we solve two equations in the microscale to obtain effective coefficients decoupling the two scales from each other. Finite-difference simulations of the macroscopic flow recover results from literature, supporting robustness of the method.Open Acces

Variational methods and its applications to computer vision

Many computer vision applications such as image segmentation can be formulated in a ''variational'' way as energy minimization problems. Unfortunately, the computational task of minimizing these energies is usually difficult as it generally involves non convex functions in a space with thousands of dimensions and often the associated combinatorial problems are NP-hard to solve. Furthermore, they are ill-posed inverse problems and therefore are extremely sensitive to perturbations (e.g. noise). For this reason in order to compute a physically reliable approximation from given noisy data, it is necessary to incorporate into the mathematical model appropriate regularizations that require complex computations. The main aim of this work is to describe variational segmentation methods that are particularly effective for curvilinear structures. Due to their complex geometry, classical regularization techniques cannot be adopted because they lead to the loss of most of low contrasted details. In contrast, the proposed method not only better preserves curvilinear structures, but also reconnects some parts that may have been disconnected by noise. Moreover, it can be easily extensible to graphs and successfully applied to different types of data such as medical imagery (i.e. vessels, hearth coronaries etc), material samples (i.e. concrete) and satellite signals (i.e. streets, rivers etc.). In particular, we will show results and performances about an implementation targeting new generation of High Performance Computing (HPC) architectures where different types of coprocessors cooperate. The involved dataset consists of approximately 200 images of cracks, captured in three different tunnels by a robotic machine designed for the European ROBO-SPECT project.Open Acces