# Analysis & Probability

## Adaptive Monte Carlo Methods for Stochastic Differential Equations

Reference Number: 2018-HW-AMS-14

Abstract: Capturing rare events is crucial for accurate risk assessment and its successful management. An example in finance is computing the probability of a large, but rare, loss from a financial portfolio. Approximating expectations involving such rare events is difficult because, when using Monte Carlo, many of the generated samples do not contribute to the final outcome and the expensive samples are effectively wasted. Adaptive sampling methods resolve this issue by spending some minimal computational effort to determine if a sample is of interest and, only if it is, the sample accuracy is then improved by spending further computational effort.  Using concepts from stochastic analysis, probably theory and numerical analysis, this project will look at applying adaptive methods to compute outputs depending on stochastic differential equations rather than simple random variables.

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## Approximating persistence times of endemic infections

Reference Number: 2018-HW-AMS-02

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: 2018-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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## Convergence to equilibrium for stochastic differential equations (SDEs) with multiple invariant measures

Reference Number: 2018-HW-Maths-55

Abstract: The ergodic theory for SDEs is a well developed branch of mathematics. Broadly speaking, such a theory, initiated in the thirties by Birkhoff, deals with the long-time behaviour of SDEs and, more specifically, with determining i) suitable conditions under which the process admits a unique invariant measure -- which, in a probabilistic context, represents the equilibirum of the process -- and ii) studying convergence to such an invariant measure. While a large body of knowledge is available when addressing the study of ergodic processes, the development of a general framework to understand problems with multiple equilibria is at a very early stage. In particular it is well known that ergodic processes will, under appropriate conditions, converge to their unique equilibrium irrespective of the initial configuration, i.e. they will tend to lose memory of the initial datum. This is a desirable property in uncertainty quantification, e.g. in the context of statistical sampling. However such a property is not, in general, satisfied by many non-synthetic systems (birds don't flock just in one direction, the microstructure of nematic crystals aligns to several possible equilibrium configurations) as the initial state does, in general, influence the asymptotic behaviour. This project will deal with the very timely problem of studying stochastic dynamics (mostly diffusion processes) with multiple equilibria. The project will be in collaboration with Imperial College London and the student taking up this research will be encouraged to travel to London for research visits.

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## Dynamic Properties of Multi-Dimensional Stochastic Processes, with Applications

Reference Number: 2018-HW-AMS-03

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: 2018-HW-Maths-19

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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## Elasticity models for image comparison

Reference Number: 2018-HW-Maths-05

Abstract: Abstract: The project will investigate nonlinear elasticity based methods for comparing two images (shapes, colours …). This involves minimizing a suitable energy functional among invertible maps taking one image onto the other. The project will thus provide a training in techniques of the calculus of variations, solid mechanics and computer vision

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## Evolution problems in non-self-adjoint spectral theory

Reference Number: 2018-HW-Maths-21

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: 2018-HW-Maths-09

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: 2018-HW-Maths-32

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: 2018-HW-Maths-10

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: 2018-HW-Maths-30

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: 2018-HW-AMS-04

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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## McKean-Vlasov SDEs: stochastic models for interacting particle systems

Reference Number: 2018-HW-Maths-16

Abstract: Many interesting systems in physics and in the applied sciences are constituted by a large number of like-particles, or agents, (e.g. individuals, animals, cells, robots) that interact with each other. Quite often, we are not interested in a detailed description of the system but rather in its collective behaviour. A rather established methodology in statistical physics and in kinetic theory is to look for simplified models that retain all the relevant characteristics of the original system, and this is how stochastic differential equations of McKean-Vlasov type arise. In this class of models, interactions (local or non-local) between agents lead to spontaneous formation of non-static, macroscopically observable patterns. Such a phenomenon goes under the name of self-organization. The recent increasing popularity of these models stems partly from the fact that they can be adapted to describe a huge range of phenomena. In economics particles are traders or companies, each of them characterized by an initial wealth, which is updated after interactions (trades). In biology applications range from bird flocking (each bird interacts with the others by aligning its velocity to the average velocity of its neighbours), to cell migration and tumor growth, mounds built by termites; further applications include models for opinion formation, rating systems, human traffic and modelling of crime-affected urban areas; a vast array of examples comes from control engineering as well. This project will be devoted to studying McKean-Vlasov SDEs; these equations are non-linear in the sense that the process depends on its own law and this renders the study of qualitative properties of the dynamics particularly challenging. We will focus on the case in which such models exhibit many invariant measures (which correspond to the formation of different possible observable patterns). The project will be in collaboration with Imperial College London and the student taking up this research will be encouraged to travel to London for research visits.

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## Modelling, Analysis, and Numerics for Stochastic Multiscale Systems

Reference Number: 2018-HW-Maths-29

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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## Multiscale diffusion processes in swarm robotics

Reference Number: 2018-HW-Maths-11

Abstract: Biological organisms have found efficient strategies to move and disperse in search of randomly located targets, like food. ​S​warm robotics ​aims at ​​developing bio-inspired ​computational​ models​ usually combined​ with machine learning techniques to ​allow robots ​to autonomously ​explore areas for tasks like search-rescue missions, monitoring, surveillance​, ​forest fire combat​ and agricultural applications, just to name a few​.A main mathematical approach for understanding the biological ​organisms ​movement or robot​ ​movement relates it to diffusion problems.​ ​This project derives macroscopic diffusion equations for the dispersion of robot ​swarms. Their pure and numerical analysis leads to efficient search strategies in applications. Diffusion equations are long-studied for biological or physical systems, such as the Keller-Segel model for cell movement in the presence of chemical cues. Robotic systems open up new possibilities of interaction, coordination and control. They lead to challenges in the mathematical analysis and the development of algorithms. In particular, the macroscopic description allows the efficient optimisation of movement rules and real-time learning and evaluation of strategies.

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## Multivariate risk processes in the presence of heavy and/or light tails

Reference Number: 2018-HW-AMS-07

Abstract: We plan to find and analyse the tail asymptotics for the ruin probabilities of multivariate insurance-reinsurance stochastic processes when the claim size distributions are either heavy- or light-tailed. We will consider various dependence structures of the components of the processes.

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## Numerical Methods for Financial Market Models

Reference Number: 2018-HW-AMS-01

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: 2018-HW-Maths-18

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.

## 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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## Quantitative Methods and Models with Applications in Finance and Actuarial Science: Valuation Techniques, Investment Strategies and Dependence Modeling

Reference Number: 2018-HW-AMS-15

Abstract:
Various stakeholders in finance and insurance—such as regulators, investors and managers—rely on quantitative analysis in their decision-making processes. This research project employs quantitative models and methods from probability theory and statistics to tackle problems that are of practical relevance in these fields. Three topics are mainly concerned.  The first topic studies numerical techniques that are useful in financial and actuarial valuation such as option pricing, capital allocation and risk aggregation etc. We aim to propose new efficient computational methods and techniques. The second topic studies investment strategies and behaviors under general risk preference with emphasis on portfolio selection, skewness preference and performance measure etc. The third topic delves into dependence modeling of risks and its applications in finance and insurance. It covers popular research questions such as model uncertainty, systemic risk, high-dimensional risk measure and worst-scenario analysis etc.

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

Reference Number: 2018-HW-Maths-28

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: 2018-HW-AMS-05

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: 2018-HW-Maths-44

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.

## Uniform in time convergence of numerical methods for Stochastic Differential Equations

Reference Number: 2018-HW-Maths-56

Abstract: As is well-known, most Stochastic Differential equations (SDEs) are not analytically solvable and for this reason numerical methods play an important role in the understanding of a large class of stochastic dynamics. Numerical schemes are, substantially, time-discrete dynamics which are supposed to stay “close” to the real solution of the SDE, i.e. they are dynamics which give a good approximation of the original SDE. An important problem is to quantify a priori “how close” the numerical method and the SDE actually are and which properties of the SDE are retained by the discretization. Many criteria have been developed to this effect. It is reasonable to expect that, the longer we run the simulation, the bigger the error between the simulated and the real dynamics – if this was not the case, we would be able to simulate the behaviour of the universe for centuries to come and take a good look into the future! The purpose of this project is to address the following question: in which cases can we simulate the dynamics in a way that the numerical error stays bounded in time? The answer will depend both on the numerical method used and on the dynamics which is being approximated. This project will lead to the formulation of novel criteria (and schemes) to establish when an SDE can be numerically integrated with an error which is uniform in time. This problem is surprisingly complex to tackle, and it requires the use of a wealth of techniques, ranging from semigroup theory to Malliavin Calculus. The student will therefore spend some time at the beginning of the project to familiarise with different techniques.

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