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Analysis & Probability

Approximating persistence times of endemic infections

Reference Number: 2017-HW-AMS-38

Abstract: The spread of infectious disease through a population may be modeled as a stochastic process (typically a continuous-time Markov chain). For infections which are able to persist in the long term (i.e. become endemic in the population), a random variable of interest is the time until eventual extinction of infection.  Programmes exist aimed at global or regional eradication of specific diseases including polio, malaria, measles, onchocerciasis and others; economic planning for such programmes could potentially be helped by good estimates of the expected time to achieve disease extinction.  For relatively simple mathematical models, the expected persistence time may be computed exactly from general Markov process theory.  For more realistic models, this approach is no longer feasible, and approximations must be sought.  This project will use recently-developed methods from statistical mechanics to approximate persistence time for a variety of infectious transmission models, and investigate the effects of disease features (e.g. length of latent period, variability of infectious period, etc) upon this persistence time.

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Computation and modelling of noise pollution in marine reserves

Reference Number: 2017-HW-Maths-17

Abstract: This research project will examine acoustic wave propagation in the marine environment to predict and estimate the extent of noise pollution in designated marine reserves from shipping lanes and other human activities such as drilling. We will develop stochastic wave equation models in collaboration with the underwater acoustics department at the School of Engineering and Physical Sciences, Heriot-Watt. These will then be analysed and fast numerical methods developed.  Depending on the interests of the PhD candidate, the topic of the PhD can be in a number of directions: modelling of noise pollution in a marine environment, mathematical analysis of stochastic wave equations, and the developments of efficient numerical methods for the solution of these.

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Degenerate Diffusion processes

Reference Number: 2017-HW-Maths-43

Abstract: It is by now well understood that the study of diffusion processes is most prolific when analytic and probabilistic techniques are combined. Indeed, diffusion processes are described at the microscopic level by  Stochastic Differential Equations (SDEs) and, at the macroscopic level, by  Partial Differential Equations (PDEs) (they most classical example being the "duality" Brownian motion/heat equation). The framework intoduced by Kolmogorov explains precisely the link between the two descriptions. The study of diffusion processes of elliptic and hypoelliptic type has by now produced a fully-fledged theory, involving several branches of mathematics: stochastic analysis, statistical mechanics, analysis of differential operators, differential geometry and control theory. One of the key steps in the development of such a theory has been the seminal paper of Hoermander (1967) and a large body of work has been dedicated for over 40 years to the study of diffusion processes under the Hoermander Condition (HC), which is a sufficient condition for hypoellipticity. Indeed, at least three Fields medallists have worked on various aspects of this topic. With this project we intend to study the properties of diffusion process which are more general than hypoelliptic; in particular, we want to relax the Hoermander condition and replace it with the so-called  UFG condition ; the main focus of the project will be the study of the long-time behaviour of Feynman-Kac semigroups (FKS) generated by UFG diffusions perturbed by both deterministic and random potentials. Ideally, this project would also explore the advantages of using degenerate processes for sampling and related numerical issues.

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Dynamic Properties of Multi-Dimensional Stochastic Processes, with Applications

Reference Number: 2017-HW-AMS-12

Abstract: We plan to develop probabilistic methods for studying long-time behaviour of complex stochastic processes with inter-dependent components. Our results will have applications in wireless communication networks, queueing systems, energy, economics and other areas.

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Efficient numerical methods for stochastic differential equations

Reference Number: 2017-HW-Maths-15

Abstract: Many applications give rise to differential equations which include some form of a time dependent noise. For example noise might be included in the model of molecular movement. These equations also arise in filtering - an important SPDE used to reconstruct data from signals. Typically these are stochastic PDEs or stochastic differential equations potentially coupled to PDEs. The aim of this project is to develop novel solution techniques and that can be used to estimate the uncertainty in computed results and to examine the efficiency and convergence of these methods. The project would develop new methods that are adaptive in space and time and may take account of different scales in the problem.

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Evolution problems in non-self-adjoint spectral theory

Reference Number: 2017-HW-Maths-37

Abstract: Operators underlying non-linear phenomena are often non-self-adjoint. As an example, linearising the equation governing the behaviour of a viscous fluid inside a rotating cylinder with axis parallel to the ground, gives rise to a highly non-self-adjoint operator with boundary and interior singularities. The evolution equation associated to this operator exhibits very unusual properties: there is a dense set of initial conditions (the eigenfunctions of the operator and any finite linear combination of them) for which there is a solution for all time, yet there is also a dense set of initial conditions for which there is no solution for any positive time. Many more examples of this sort have been found recently. The goal of this PhD project will be to closely examine specific classes of non-self-adjoint operators for which the associated evolution problem exhibits a wild behaviour.

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Finite elements for wave propagation: Analysis, fast methods and concert halls

Reference Number: 2017-HW-Maths-26

Abstract: This project concerns considers accurate and efficient numerical methods for sound propagation, often steered by hard analysis. Some key challenges of wave computations arise in the context of an instrument placed in a concert hall: High frequencies, nonlinear wave propagation and boundary conditions, complex domains and sound emission.  On the other hand, detailed models to simulate music instruments have been investigated in numerical analysis, from violins to a grand piano. Their simulations of classical music from first principles are truly impressive, and in the longer-term we hope to place these models in realistic surroundings.
Some directions: At high frequencies naive numerical methods require a very fine mesh to capture the rapidly oscillating sound pressure. Tools from harmonic analysis and PDE give rise to new regularity estimates and efficient adaptive mesh refinements. Detailed models of instruments (or other sound sources like tires or high-speed trains) give rise to realistic simulations of sound, based on mathematically challenging multi-physics problems. 

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Formation in finite time of singularities in moving boundary problems

Reference Number: 2017-HW-Maths-31

Abstract: Moving boundary problems are nonlinear PDE problems set on a domain evolving in time, the moving boundary being part of the unknowns of the problem. It can either be one-phase (the moving boundary being a free boundary) or two-phase (the moving boundary being the interface between the two phases). The project will concentrate on establishing (by proving) some situations where finite time-singularity happens, for either the PDE or the domain (with the formation of a cusp singularity for instance).

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Fractional reaction-diffusion equations: Analysis, mathematical biology and computation

Reference Number: 2017-HW-Maths-25

Abstract: Fractional diffusion equations have attracted much recent interest, as they describe diffusion in the presence of long jumps. Analytically, the Laplace operator of the heat equation is raised to a non-integer power, resulting in a nonlocal operator; probabilistically, Brownian motion is replaced by a Levy process. This project studies either the pure analysis and  efficient numerical approximation of nonlinear reaction-diffusion equations for superdiffusing particles, as they arise in applications from the movement of cells or bacteria (chemotaxis), the spread of diseases, or in social networks.  Basic questions have only recently started to be addressed: Can we rigorously derive such equations from microscopic descriptions for the movement of bacteria, cells or particles? Are the resulting equations well-posed for all positive times, or do solutions blow up? Can we rigorously and efficiently compute the solutions, ideally with provable error bounds? Are there interesting solutions (traveling waves, pattern formation), or can we make interesting predictions for physicists or biologists?

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From Photosynthesis to Solar Cells: investigating new theoretical and computational directions

Reference Number: 2017-HW-Maths-27

Abstract: We begin with investigating the basic quantum mechanical principles such as (classical and quantum) information theory and entropy, thermodynamics of irreversible computations, superposition, locality, entanglement, and quantum gates, infinite dimensional analysis (Hilbert Spaces), which all are fundamental in the development of quantum inspired formulations.
Equipped with these tools, we will look into a quantum-like description for photosynthesis. Here, quantum-like means that the mathematical structure is different from conventional quantum mechanics. We will investigate a quantum network formulation for electron transport in organic molecules and describe the photosynthesis by a quantum channel representation. A question of general interest then is whether and how we can transfer these ideas to solar cells to ultimately improve the performance.

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Heavy-tailed distributions in stochastic networks

Reference Number: 2017-HW-AMS-02

Abstract: Distributions for which all exponential moments are infinite are called heavy-tailed. In particular, power-tail, log-normal and some Weibull distributions are heavy-tailed.  Motivated by applications in queueing theory, we will study rare events in stochastic networks with service/transmission times having a heavy-tailed distribution.

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Modelling, Analysis, and Numerics for Stochastic Multiscale Systems

Reference Number: 2017-HW-Maths-28

Abstract: The starting point will be to gain a basic understanding of the appearance and emergence of randomness in seemingly deterministic systems. To this end, we will make use of Statistical Mechanics, Analysis, PDE theory, Thermodynamics, and numerical discretisation strategies. Key problems, that we will investigate, are how to reliably coarse grain non-equilibrium systems and to develop systematic, analytical and numerical methods to solve stochastic homogenization problems which arise in science, engineering, and data science.

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Numerical Methods for Financial Market Models

Reference Number: 2017-HW-AMS-03

Abstract: Many existing models for the evolution of financial and economic variables such as interest rates, inflation and so forth have no known closed-form solution. In order to deal with such models, e.g. for pricing and risk management of financial derivatives, it is therefore of fundamental importance to design numerical methods that are highly accurate, fast and robust. This project will apply methods from stochastic analysis and probability theory to models of financial markets to enhance the understanding of their stochastic properties, and to design high-quality fast methods for their numerical treatment.

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Numerical simulation and neural modelling

Reference Number: 2017-HW-Maths-16

Abstract: Neural field equations are being used more and more to extract data from real reading - for example from fMRI scans. To be of use we need fast and convergent algorithms for their numerical simulation. This project would examine the stochastic neural field and how to compute structures, such as waves, through the field.
At the other end of the scale, with colleagues in biology we have the opportunity to take a large amount of cell movement data and to develop models for molecular movement and interactions. The project would apply new techniques for estimating parameters and could be used to inform new biological theories and experiments.

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Optimal Coupling and Rates of Convergence to Stationarity for Markov Chains, with Applications

Abstract: Consider two or more Markov chains on a finite or countable state space, each starting from a different state but evolving using the same transition probability matrix. This project will study how such Markov chains can be constructed (coupled) in an optimal way, for several different notions of optimality, and how knowledge about the time until the coalescence of trajectories may be applied in various settings.

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Parameterisations of almost flat sets

Abstract: In this project, we will investigate how different properties of the multi-scale geometry of a set ensure it has ``nice" parameterisations. For example, suppose we are given a set that we know is locally quite flat in the sense that, inside every ball centred on the set, it is concentrated around a d-dimensional plane: can we contain this set in a ``nice" surface?

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Probability Models of Cancer Growth

Reference Number: ACM

Abstract: Biology is a fast growing area for applications of probability. Since still in its infancy, there are many unexplored areas and open problems. In particular, there is a great interest in stochastic models of cancers. This PhD project would focus on understanding the most basic and fundamental models of tumor progression. These models include branching processes, other models borrowed from population genetics, or spatial Poisson processes. The work has a light numerical aspect to it, but would focus more on finding exact solutions, and establishing limit theorems. No knowledge of biology is required.

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Quantum inspired mathematical formulation for decision-making: computing and algorithms

Reference Number: 2017-HW-Maths-29

Abstract: We begin with investigating the basic quantum mechanical principles such as (classical and quantum) information theory and entropy, thermodynamics of irreversible computations, superposition, locality, entanglement, and quantum gates, infinite dimensional analysis (Hilbert Spaces), which all are fundamental in the development of quantum inspired formulations. Equipped with these tools, we will look into a quantum based description for decision making in two-player games. Here, quantum based means that the mathematical structure is different from conventional quantum mechanics. We will investigate decision processes in games of Prisoners Dilemma type. These processes are gaining increasing interest, since classical (Kolmogorov) probability or classical quantum mechanics seem not to be consistent with statistical data obtained in cognitive experiments.

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Random graphs and networks: limits, approximations, and applications

Reference Number: 2017-HW-AMS-13

Abstract: Random graphs and random networks are used as models in many diverse applications across physics, biology, engineering, computing and communications, among other areas.  The aim of this project is to study the asymptotic structure of these random objects.  What behaviour do we observe as the size of the network grows?  Can we approximate a real-world network by its asymptotic counterpart?  If so, can we quantify the error in such an approximation?

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Singular stochastic dispersive dynamics

Abstract: Nonlinear dispersive PDEs such as the nonlinear Schrodinger equations appear naturally as a model describing wave propagation in various branches of physics and engineering.  Since these equations appear in nature, they are susceptible to random noise, for example, through observations and storage process. As such, it is natural to study nonlinear dispersive PDEs with random initial data and/or random forcing.  In this Ph.D. project, we study dispersive PDEs from both deterministic and stochastic viewpoints using harmonic analysis and probability theory and try to answer some of the fundamental questions such as well-posedness, long-time behaviour and singularity formation.

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Solvability of the Neuman and regularity boundary value problems for elliptic PDEs with complex coefficients.

Abstract: The main project objective is understanding of well-posedness (existence, uniqueness and stability of solutions) of elliptic and parabolic PDEs (the initial focus will be for the elliptic case, later parabolic PDEs can be looked at). The focus will be on PDEs with complex coefficients (which can also be thought as the case of a special skew-symmetric system with real coefficients by writing separately the PDEs for the real and imaginary part).   There has been a recent breakthrough in the regularity theory for elliptic PDEs with complex coefficients via harmonic analysis. In particular, a replacement for Giorgi-Nash-Moser interior regularity has been found (via the concept of p-ellipticity) which serves as a weaker (but viable) replacement in the complex case. This so far has only been developed for the Dirichlet boundary value problem (which is the easiest one to consider). The regularity and Neumann problems require new ideas which will be developed in this project.

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Stochastic Modelling of Biological Systems

Reference Number: ACM

Abstract: Due to recent experimental progress, the field of mathematical biology is rapidly growing. There are plenty of biological systems where mathematical models and analysis are needed. Closely interacting with experimentalists, the PhD candidate would formulate and analyze models of cancer progression, virus dynamics, bacterial evolution, and possibly other systems related to molecular motor motion or the origins of life. The project starts with first building and exploring simple model systems, and continues with their study by computer simulations and analytical methods. Knowledge of the biological background is not necessary at the beginning, but one will eventually learn some biology in order to do relevant research.

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Stochastically forced compressible fluid flows

Reference Number: 2017-HW-Maths-46

Abstract: The dynamics of liquids and gases can be modelled by systems of partial di fferential equations (PDEs) describing the balance of mass, momentum and energy in the fluid flow. In the last years their was an increasing interest of random influences on their solutions. Stochastic partial differential equations (SPDEs) for viscous incompressible fluids highly attracted the interest of both analysts and probabilists. In contrast to this is the analysis of stochastically forced compressible fluid flows still very limited (with only a few rigorous results available) and shall be investigated in this project. It combines methods for PDEs with infi nite dimensional stochastic analysis. Questions of interest are existence and qualitative properties of solutions to the underlying SPDE, their long-time behavior as well as numerical approximations.

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