Theses - Cambridge Centre for Analysis (CCA)
Permanent URI for this collection
Browse
Recent Submissions
Item Open Access Asymptotically cylindrical Calabi–Yau and special Lagrangian geometry(2017-04-13) Talbot, Timothy JamesWe study asymptotically cylindrical Calabi–Yau manifolds and their asymptotically cylindrical special Lagrangian submanifolds. As a prototype problem, we also consider an extension of Hodge theory to general asymptotically cylindrical manifolds. For our study of asymptotically cylindrical Calabi–Yau manifolds, we restrict to complex dimension three. We regard a Calabi–Yau structure as a pair of closed forms ($\Omega$, $\omega$); the assumption that the structure is asymptotically cylindrical gives an asymptotic condition on ($\Omega$, $\omega$). Regarding the Riemannian products of Calabi-Yau threefolds with S$^{1}$ as G$_{2}$ manifolds, we show that the asymptotically cylindrical deformations of a Calabi–Yau structure (with possibly varying asymptotic limit) are unobstructed. Locally, the spaces of deformations are given by appropriate spaces of harmonic forms. We then show that we can glue asymptotically cylindrical Calabi–Yau manifolds, and that if we do so the “gluing map” of moduli spaces is essentially a local diffeomorphism. In particular, it is an open mapping. In the case of asymptotically cylindrical special Lagrangian submanifolds, we no longer explicitly restrict to dimension three; we assume only that we have a gluing theorem for Calabi–Yau manifolds of the kind obtained in dimension three. McLean and others have constructed deformation spaces of special Lagrangian submanifolds; we show that gluing of asymptotically cylindrical special Lagrangian submanifolds is again unobstructed. As in the Calabi–Yau case, we can define a “gluing map” and this map is a local diffeomorphism of moduli spaces. In both cases, the local diffeomorphism property gives a “local Mayer–Vietoris principle” for deformations. In the special Lagrangian case, the linearisation of the “ungluing” map so defined is just the map of harmonic forms induced by Hodge theory from the natural map of de Rham cohomology; in the Calabi–Yau case it is only slightly more involved.Item Open Access Simulation of Diffusion Limited Aggregation Models and Related ResultsBell, JamesThis thesis is in three parts. All parts are motivated by a desire to gain a better understanding of models of the phenomenon of two-dimensional diffusion limited aggregation, henceforth DLA. The first part proves some generalisations of results relating to Hastings-Levitov with α = 0, henceforth HL(0), another two-dimensional growth process. The second part is a study of numerical algorithms for simulating off-grid DLA. The third part describes and reports on some numerical experiments on multiple models of DLA. Part I provides a generalization of the concept of disturbance flows and of the coalescing Brownian flow, also known as the Brownian web, proving facts about the convergence of the former to the latter and about their time- reversals. This work was motivated as an attempt to generalize known results about the harmonic flow of HL(0) to the case of HL(2), which is supposed to be a model for DLA. Part II provides the first rigorous analysis of the asymptotic runtimes of four different previously published algorithms for simulating off-grid DLA. A variation on one of these algorithms, incorporating an improvement from another source and a trick new to this work, is implemented in code, with the runtimes comparing favourably to previous work. The runtime of this algorithm, like that of the algorithm it is based on, is Õ(n), which is optimal. Part III is a report on experiments testing whether or not off-grid DLA, HL(2) and noise-reduced DLA all have the same limiting shape in the many particle limit. It also contains a heuristic discussion of whether regularized HL can provide a good model for DLA. The results and heuristics indicate that regularizing HL with slit particles is not a promising way to simulate DLA. However, HL with circular particles, off-grid DLA and noise-reduced DLA are found to be in agreement.Item Open Access Consistency of nonparametric Bayesian methods for two statistical inverse problems arising from partial differential equations(2020-07-17) Abraham, Luke Kweku William; Abraham, Luke Kweku William [0000-0001-5243-6998]Partial differential equations (PDEs) govern many natural phenomena. When trying to understand the parameters driving these phenomena, we must be aware of the inevitable errors in our measurements; in statistical inverse problems these measurement errors are modelled by statistical noise. One approach to recovering the PDE coefficients governing such statistical inverse problems is through Bayesian methodology. This thesis investigates the theoretical performance of the Bayesian approach in two particular cases. The first model considered is the advection-diffusion equation. Kolmogorov’s equations link this partial differential equation to a corresponding (time-homogeneous) stochastic differential equation, in which a diffusion process flows according to a ‘drift function’ and is buffeted by a Brownian motion effect of spatially varying magnitude; this diffusion formulation forms the focus herein. Assuming the diffusion coefficient (the magnitude of the Brownian effect) is given, this thesis considers the problem of recovering the drift function from observations of the diffusion at discrete time intervals. Chapter 2 gives explicit conditions on priors under which the corresponding Bayesian posteriors provably contract in $L^2$ distance, as data is collected, around the true drift function, at the frequentist minimax rate (up to logarithmic factors) over periodic Besov smoothness classes. These conditions are verified for some natural nonparametric priors, some of which are shown to adapt to an unknown smoothness parameter. The results are given in the high-frequency regime, where the diffusion is observed to a later time horizon and at ever closer intervals, but in fact the minimax rate (again up to logarithmic factors) is also attained in the low-frequency regime, where the intervals between samples remain fixed. This yields the first drift estimator robust to the sampling regime. The second model considered is the Calderón problem. This is the mathematical formulation of electrical impedance tomography, in which electrodes are attached to a patient’s skin and used to apply voltages and record the corresponding current fluxes. The current flux corresponds to the Neumann data for the solution to a PDE, governed by an interior ‘conductivity parameter’, in which the voltage gives the Dirichlet boundary values. Varying the applied voltage, we consider observing the ‘Dirichlet-to-Neumann map’, and attempt to recover the interior conductivity. The data considered in Chapter 3 consists of the Dirichlet-to-Neumann map corrupted by additive Gaussian noise. A prior is exhibited for which the posterior mean statistically converges to the true conductivity (as the noise level is taken to 0) at a near-optimal rate. The introductory chapter outlines the minimax framework by which the posteriors are judged, and provides the background material relevant to this thesis. Of particular interest may be the included proof, in an general inverse problem setting, of natural conditions under which the consistency of the posterior mean can be guaranteed.Item Open Access On Gaussian Multiplicative Chaos(2019-07-20) Wong, Mo DickGaussian multiplicative chaos was first constructed in Kahane's seminal paper in 1985 in an attempt to provide a mathematical foundation for Kolmogorov-Obukhov-Mandelbrot theory of energy dissipation in developed turbulence. It has attracted a lot of attentions from the mathematics community in the last decade, playing a pivotal role in the probabilistic formulation of Liouville conformal field theory, as well as showing up in different branches of mathematics such as analytic number theory where it describes the statistical behaviour of the Riemann zeta function on the critical line. This thesis explores the theory of Gaussian multiplicative chaos in three different directions. We commence with a new connection with random matrix theory, showing that for large Hermitian matrices sampled from the one-cut-regular unitary ensemble, the absolute powers of the characteristic polynomial, when suitably normalised, converge in distribution to multiplicative chaos on the support of the limiting spectral distribution as the size of the matrix goes to infinity, and the limit is independent of the choice of the potential function. This is part of an ongoing programme of establishing Gaussian multiplicative chaos as a universal limit object in probability theory. Next, we consider Gaussian multiplicative chaos in the context of Liouville conformal field theory and study the fusion estimate of the Liouville correlation function. More precisely, we derive the exact asymptotics for the Liouville four-point correlation when two points are merging and express the leading order coefficient in terms of DOZZ constants from the three-point correlation function. Our result is consistent with predictions from conformal bootstrap in theoretical physics, and has a geometric interpretation of surfaces being glued together, as hinted by the bootstrap equation. Finally, we study the right tail of the mass of Gaussian multiplicative chaos and establish a formula for the leading order asymptotics under mild assumptions on the underlying log-correlated Gaussian field. The tail exponent satisfies a universal power-law profile, while the leading order coefficient can be described by the product of two constants, one capturing the dependence on the test set and any non-stationarity, and the other one encoding the universal properties of multiplicative chaos. This may be seen as a first step in understanding the full distributional properties of Gaussian multiplicative chaos.Item Open Access Optimal estimation in high-dimensional and nonparametric models(2019-05-18) Baldin, NikolayMinimax optimality is a key property of an estimation procedure in statistical modelling. This thesis looks at several problems in high-dimensional and nonparametric statistics and proposes novel estimation procedures. It then provides statistical guarantees on the performance of these methods and establishes whether those are computationally tractable. In the first chapter, a new estimator for the volume of a convex set is proposed. The estimator is minimax optimal and also efficient non-asymptotically: it is nearly unbiased with minimal variance among all unbiased oracle-type estimators. Our approach is based on a Poisson point process model and as an ingredient, we prove that the convex hull is a sufficient and complete statistic. No hypotheses on the boundary of the convex set are imposed. In a numerical study, we show that the estimator outperforms earlier estimators for the volume. In addition, an improved set estimator for the convex body itself is proposed. The second chapter extends the results of the first chapter and develops a unified framework for estimating the volume of a set in $R^d$ based on observations of points uniformly distributed over the set. The framework applies to all classes of sets satisfying one simple axiom: a class is assumed to be intersection stable. No further hypotheses on the boundary of the set are imposed; in particular, the class of convex sets and the class of weakly-convex sets are covered by the framework. We introduce the so-called wrapping hull, a generalization of the convex hull, and prove that it is a sufficient and complete statistic. The proposed estimator of the volume is simply the volume of the wrapping hull scaled with an appropriate factor. It is shown to be consistent for all classes of sets satisfying the axiom and mimics an unbiased estimator with uniformly minimal variance. The construction and proofs hinge upon an interplay between probabilistic and geometric arguments. The tractability of the framework is numerically confirmed in a variety of examples. The third chapter considers the problem of link prediction, based on partial observation of a large network, and on side information associated to its vertices. The generative model is formulated as a matrix logistic regression. The performance of the model is analysed in a high-dimensional regime under a structural assumption. The minimax rate for the Frobenius-norm risk is established and a combinatorial estimator based on the penalised maximum likelihood approach is shown to achieve it. Furthermore, it is shown that this rate cannot be attained by any (randomised) polynomial-time algorithm under a computational complexity assumption. The trade-off between computational efficiency and statistical optimality is discussed throughout the thesis. For estimating the volume of a set from the class of convex or weakly-convex sets in high dimensions, we propose minimax optimal estimators in the first and second chapters. However, they cannot be computed using a polynomial-time algorithm in dimensions higher than three. Analogously, the proposed minimax optimal estimator for a prediction task in the matrix logistic regression problem in the third chapter cannot be computed in polynomial time. The third chapter further identifies a computational lower bound in the regression problem, thereby revealing the gap between the best possible rate of convergence of a polynomial-time algorithm and the minimax optimal rate.Item Open Access Charged scalar fields on Black Hole space-times(2019-05-18) Van de Moortel, Maxime Claude RobertThe goal of this thesis is to study charged Black Holes in the presence of charged matter. To do so, we investigate the behaviour of spherically symmetric solutions of the Einstein-Maxwell-Klein-Gordon equations, which model the interaction of a charged scalar field with the electromagnetic field originating from the Black Hole charge. The particularity of this model is to putatively admit charged one-ended Black holes with a Cauchy horizon, and thus provides a framework to study simultaneously charged gravitational collapse and the Strong Cosmic Censorship conjecture. The latter problem is related to the question of Determinism of General Relativity, and roughly states that the maximal development of admissible initial data is inextendible. This question is intimately connected to the geometry of the Black Hole interior, which is studied in the first chapter of the present thesis. We prove that perturbed charged Black Holes form a Cauchy horizon which admits generically a singularity. This singularity in turn forms an obstruction to extending the maximal development. To obtain this result, we undertake an asymptotic analysis of the scalar field in the interior of the Black Hole, assuming its exterior region settles towards a stationary solution at a time decay rate that is expected by numeric and heuristic works. In the second chapter of this thesis, we retrieve these time decay rates for weakly charged scalar field on a fixed Reissner-Nordstrom Black Hole exterior. The result provides a proof of the (gravity-uncoupled) stability of Reissner-Nordstrom Black Hole exterior against small charged perturbations, which should also be considered as the first step towards the construction of one-ended charged Black Holes with a Cauchy horizon.Item Open Access Functional inequalities in quantum information theory(2019-07-22) Rouzé, CambyseFunctional inequalities constitute a very powerful toolkit in studying various problems arising in classical information theory, statistics and many-body systems. Extensions of these tools to the noncommutative setting have been introduced in the beginning of the 90's in order to study the asymptotic properties of certain quantum Markovian evolutions. In this thesis, we study various extensions and problems arising from the specific noncommutative nature of such processes. The first logarithmic Sobolev inequality to be proved, due to Gross, was for the Ornstein Uhlenbeck semigroup, that is the Brownian motion with friction on the real line. The generalization of this result to the quantum Ornstein Uhlenbeck semigroup was found very recently by Carlen and Maas, and de Palma and Huber by means of different techniques. The latter proof consists of a quantum generalization of the so-called entropy power inequality. Here, we consider another possible version of the entropy power inequality and use it to derive asymptotic properties of the frictionless quantum Brownian motion. The proof of Carlen and Maas discussed in the previous paragraph relies on their new quantum extension of the classical notion of displacement convexity. This is classically known to imply most of the usual functional inequalities such as the modified logarithmic Sobolev inequality and Poincaré's inequality. Here, we further study the framework introduced by Carlen and Maas. In particular, we show how displacement convexity implies quantum functional and transportation cost inequalities. The latter are then used to derive certain concentration inequalities of quantum states in the spirit of Bobkov and Goetze. These concentration inequalities are used in order to derive finite sample size bounds for the task of quantum parameter estimation. The main advantage of classical logarithmic Sobolev inequalities over other methods resides in their tensorization property: the strong log-Sobolev constant of the product of independent Markovian evolutions is equal to the maximum over the set of strong log-Sobolev constants of the individual evolutions. However, this property is strongly believed to fail in the non-commutative case, due to the non-multiplicativity of noncommutative Lp to Lq norms. In this thesis, we show tensorization of the logarithmic Sobolev constants for the simplest quantum Markov semigroup, namely the generalized depolarizing semigroup. Moreover, we consider a new general method to overcome the issue of tensorization for general primitive quantum Markov semigroups by looking at their contractivity properties under the completely bounded Lp to Lq norms. This method was first investigated in the restricted case of unital semigroups by Beigi and King. Noncommutative functional inequalities considered in the present literature only deal with primitive quantum Markovian semigroups which model memoryless irreversible dynamics converging to a specific faithful state. However, quantum Markov semigroups can in general display a much richer behavior referred to as decoherence: In particular, under some mild conditions, any such semigroup is known to converge to an algebra of observables which effectively evolve unitarily. Here, we introduce the concept of a decoherence-free logarithmic Sobolev inequality, and the related notion of hypercontractivity of the associated evolution, to study the decoherence rate of non-primitive quantum Markov semigroups. Moreover, we utilize the transference method recently introduced by Gao, Junge and LaRacuente, in order to find decoherence times associated to a class of decoherent Markovian evolutions of great importance in the field of quantum error protection, namely collective decoherence semigroups. Finally, we develop the notion of quantum reverse hypercontractivity, first introduced by Cubitt, Kastoryano, Montanaro and Temme in the unital case, and apply it in conjunction with the tensorization of the modified logarithmic Sobolev inequality for the generalized depolarizing semigroup in order to find strong converse rates in quantum hypothesis testing and for the classical capacity of classical-quantum channels. Moreover, the transference method also allows us to find strong converse bounds on the various capacities of quantum Markovian evolutions.Item Open Access Asymptotic Behaviour and Derivation of Mean Field Models(2019-04-01) Holding, Thomas JamesThis thesis studies various problems related to the asymptotic behaviour and derivation of mean field models from systems of many particles. Chapter 1 introduces mean field models and their derivation, and then summarises the following chapters of this thesis. Chapters 2, 3 and 4 directly study systems composed of many particles. In Chapter 2 we prove quantitative propagation of chaos for systems of interacting SDEs with interaction kernels that are merely Hölder continuous (the usual assumption being Lipschitz). On the way we prove the existence of differentiable stochastic flows for a class of degenerate SDEs with rough coefficients and a uniform law of large numbers for SDEs. Chapters 3 and 4 study the asymptotic behaviour of the Arrow-Hurwicz-Uzawa gradient method, which is a dynamical system for locating saddle points of concave-convex functions. This method is widely used in distributed optimisation over networks, for example in power systems and in rate control in communication networks. Chapter 3 gives an exact characterisation of the limiting solutions of the gradient method on the full space for arbitrary concave-convex functions. In Chapter 4 we extend this result to the subgradient method where the dynamics of the gradient method are restricted to an arbitrary convex set. Chapters 5, 6 and 7 study the stability of mean field models. Chapters 5 and 6 prove an instability criterion for non-monotone equilibria of the Vlasov-Maxwell system. In Chapter 5 we study a related problem in approximation of the spectra of families of unbounded self adjoint operators. In Chapter 6 we show how the instability problem for Vlasov-Maxwell can be reduced to this spectral problem. In Chapter 7 we give a proof of well-posedness of a class of solutions to the Vlasov-Poisson system with unbounded spatial density. Chapters 8 and 9 change track and study the dynamics of a solute in a fluid background. In Chapter 8 we study a simple model for this phenomena, the kinetic Fokker-Planck equation, and show contraction of its semi-group in the Wasserstein distance when the spatial variable lies on the torus. Chapter 9 studies a more complex model of passive transport of a solute under a large and highly oscillatory fluid field. We prove a homogenisation result showing convergence to an effective diffusion equation for the transported solute profile.Item Open Access Anisotropic variational models and PDEs for inverse imaging problems(2019-10-26) Parisotto, Simone; Parisotto, Simone [0000-0003-0865-0289]In this thesis we study new anisotropic variational regularisers and partial differential equations (PDEs) for solving inverse imaging problems that arise in a variety of real-world applications. Firstly, we introduce a new anisotropic higher-order total directional variation regulariser. We describe both the theoretical and the numerical details for its use within a variational formulation for solving inverse problems and give examples for the reconstruction of noisy images and videos, image zooming and the interpolation of scattered surface data. Secondly, we focus on a non-symmetric drift-diffusion equation, called osmosis. We propose an efficient numerical implementation of the osmosis equation, based on alternate directions and operator splitting techniques. We study their scale-space properties and show their efficiency in processing large images. Moreover, we generalise the osmosis equation to accommodate suitable directional information: this modification turns out to be useful to correct for the well-known blurring artefacts the original osmosis model introduces when applied to shadow removal in images. Last but not least, we explore applications of variational models and PDEs to cultural heritage conservation. We develop a new non-invasive technique that uses multi-modal imaging for detecting sub-superficial defects in fresco walls at sub-millimetre precision. We correct light-inhomogeneities in these imaging measurements that are due to measurement errors via osmosis filtering, in particular making use of the efficient computational schemes that we introduced before for dealing with the large-scale nature of these measurements. Finally, we propose a semi-supervised workflow for the detection and inpainting of defects in damaged illuminated manuscripts.Item Open Access Relaxation to equilibrium for kinetic Fokker-Planck equation(2019-02-01) Piazzoli, DavideWe want to study long-time behaviour of solutions $f_t$ of kinetic Fokker-Planck equation in $\mathbb{R}^d$, namely their convergence towards equilibrium $f_\infty$ in the form \[ \textrm{d}(f_t,f_\infty)\leq C_1 e^{-C_2 t}\textrm{d}(f_0,\mu) \] for appropriate distances $\textit{d}$ and constants $C_1 \geq 1$, $C_2>0$. In Section 1 we provide an introduction and motivation for the equation, together with the setting of {Villani, Hypocoercivity} which will be useful in Section 2. In Section 2 we will review the monograph {Villani, Hypocoercivity}, where such convergence is proved, for $h=f/\mu$, in $H^1 (\mu)$ and $H_\mu +I_\mu$, that is, the sum of relative entropy and Fisher information. Here results are stated in terms of general operators $\partial_t +A^*A+B=0$, and commutation conditions on $A$ and $B$ are to be imposed. In Section 3 we shall take into consideration the work by Monmarch\'{e} {Monmarche, Generalized Γ calculus} in which such convergence is established by rephrasing some concepts in term of $\Gamma$-calculus: with respect to {Villani, Hypocoercivity} there is no need for regularization along the semigroup since the functional taken into account is a modified $H+I$ that at initial time only takes entropy into account, and the argument turns out to be shorter. Also, the convergence rate is $e^{-Ct(1-e^{-t})^2}$ instead of $C_1 e^{-C_2 t}$. However it turns out, as in {Villani, Hypocoercivity}, that for this case it is strictly needed to have a pointwise bound on $D^2 U$, where $U$ is the confinement potential. A drawback of this method with respect to {Villani, Hypocoercivity} is that, in a more general setting than kinetic Fokker-Planck equation, stronger commutation assumptions are required, which imply that the diffusion matrix is basically required to be constant. On this work a specific analysis was carried out, simplifying the proof for our Fokker-Planck case and finding explicit and improved expressions for convergence constants. The same author in {Monmarche, chaos kinetic particles}, which is the subject of Section 4, addresses a Vlasov-Fokker-Planck equation with a potential that generalizes $U$ and the related particle system. Chaos propagation in $W_2$, the $2$-Wasserstein distance, is proved, namely $W_2(f_t^{(1,N)},f_t)\leq CN^{-\epsilon}$. This leads to both Wasserstein and $L^1$ hypocoercivity, however dependence of the right hand side from the initial data is not linear as wished.Item Open Access Deterministic and Stochastic Approaches to Relaxation to Equilibrium for Particle Systems(2019-01-26) Evans, Josephine Angela HollyThis work is about convergence to equilibrium problems for equations coming from kinetic theory. The bulk of the work is about Hypocoercivity. Hypocoercivity is the phenomenon when a semigroup shows exponentially relaxation towards equilibrium without the corresponding coercivity (dissipativity) inequality on the Dirichlet form in the natural space, i.e. a lack of contractivity. In this work we look at showing hypocoercivity in weak measure distances, and using probabilistic techniques. First we review the history of convergence to equilibrium for kinetic equations, particularly for spatially inhomogeneous kinetic theory (Boltzmann and Fokker-Planck equations) which motivates hypocoercivity. We also review the existing work on showing hypocoercivity using probabilistic techniques. We then present three different ways of showing hypocoercivity using stochastic tools. First we study the kinetic Fokker-Planck equation on the torus. We give two different coupling strategies to show convergence in Wasserstein distance, $W_2$. The first relies on explicitly solving the stochastic differential equation. In the second we couple the driving Brownian motions of two solutions with different initial data, in a well chosen way, to show convergence. Next we look at a classical tool to show convergence to equilibrium for Markov processes, Harris's theorem. We use this to show quantitative convergence to equilibrium for three Markov jump processes coming from kinetic theory: the linear relaxation/BGK equation, the linear Boltzmann equation, and a jump process which is similar to the kinetic Fokker-Planck equation. We show convergence to equilibrium for these equations in total variation or weighted total variation norms. Lastly, we revisit a version of Harris's theorem in Wasserstein distance due to Hairer and Mattingly and use this to show quantitative hypocoercivity for the kinetic Fokker-Planck equation with a confining potential via Malliavin calculus. We also look at showing hypocoercivity in relative entropy. In his seminal work work on hypocoercivity Villani obtained results on hypocoercivity in relative entropy for the kinetic Fokker-Planck equation. We review this and subsequent work on hypocoercivity in relative entropy which is restricted to diffusions. We show entropic hypocoercivity for the linear relaxation Boltzmann equation on the torus which is a non-local collision equation. Here we can work around issues arising from the fact that the equation is not in the H\"{o}rmander sum of squares form used by Villani, by carefully modulating the entropy with hydrodynamical quantities. We also briefly review the work of others to show a similar result for a close to quadratic confining potential and then show hypocoercivity for the linear Boltzmann equation with close to quadratic confining potential using similar techniques. We also look at convergence to equilibrium for Kac's model coupled to a non-equilibrium thermostat. Here the equation is directly coercive rather than hypocoercive. We show existence and uniqueness of a steady state for this model. We then show that the solution will converge exponentially fast towards this steady state both in the GTW metric (a weak measure distance based on Fourier transforms) and in $W_2$. We study how these metrics behave with the dimension of the state space in order to get rates of convergence for the first marginal which are uniform in the number of particles.Item Open Access On fundamental computational barriers in the mathematics of information(2018-10-20) Bastounis, Alexander JamesThis thesis is about computational theory in the setting of the mathematics of information. The first goal is to demonstrate that many commonly considered problems in optimisation theory cannot be solved with an algorithm if the input data is only known up to an arbitrarily small error (modelling the fact that most real numbers are not expressible to infinite precision with a floating point based computational device). This includes computing the minimisers to basis pursuit, linear programming, lasso and image deblurring as well as finding an optimal neural network given training data. These results are somewhat paradoxical given the success that existing algorithms exhibit when tackling these problems with real world datasets and a substantial portion of this thesis is dedicated to explaining the apparent disparity, particularly in the context of compressed sensing. To do so requires the introduction of a variety of new concepts, including that of a breakdown epsilon, which may have broader applicability to computational problems outside of the ones central to this thesis. We conclude with a discussion on future research directions opened up by this work.Item Open Access Asymptotic theory for Bayesian nonparametric procedures in inverse problems(2015-07-18) Ray, KolyanThe main goal of this thesis is to investigate the frequentist asymptotic properties of nonparametric Bayesian procedures in inverse problems and the Gaussian white noise model. In the first part, we study the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. This rate provides a quantitative measure of the quality of statistical estimation of the procedure. A theorem is proved in a general Hilbert space setting under approximation-theoretic assumptions on the prior. The result is applied to non-conjugate priors, notably sieve and wavelet series priors, as well as in the conjugate setting. In the mildly ill-posed setting, minimax optimal rates are obtained, with sieve priors being rate adaptive over Sobolev classes. In the severely ill-posed setting, oversmoothing the prior yields minimax rates. Previously established results in the conjugate setting are obtained using this method. Examples of applications include deconvolution, recovering the initial condition in the heat equation and the Radon transform. In the second part of this thesis, we investigate Bernstein--von Mises type results for adaptive nonparametric Bayesian procedures in both the Gaussian white noise model and the mildly ill-posed inverse setting. The Bernstein--von Mises theorem details the asymptotic behaviour of the posterior distribution and provides a frequentist justification for the Bayesian approach to uncertainty quantification. We establish weak Bernstein--von Mises theorems in both a Hilbert space and multiscale setting, which have applications in $L^2$ and $L^\infty$ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets using a Bayesian approach. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the different geometries involved.Item Open Access An Optimisation-Based Approach to FKPP-Type Equations(2018-07-21) Driver, David Philip; Driver, David Philip [0000-0002-4159-8120]In this thesis, we study a class of reaction-diffusion equations of the form $\frac{\partial u}{\partial t} = \mathcal{L}u + \phi u - \tfrac{1}{k} u^{k+1}$ where $\mathcal{L}$ is the stochastic generator of a Markov process, $\phi$ is a function of the space variables and $k\in \mathbb{R}\backslash\{0\}$. An important example, in the case when $k>0$, is equations of the FKPP-type. We also give an example from the theory of utility maximisation problems when such equations arise and in this case $k<0$. We introduce a new representation, for the solution of the equation, as the optimal value of an optimal control problem. We also give a second representation which can be seen as a dual problem to the first optimisation problem. We note that this is a new type of dual problem and we compare it to the standard Lagrangian dual formulation. By choosing controls in the optimisation problems we obtain upper and lower bounds on the solution to the PDE. We use these bounds to study the speed of the wave front of the PDE in the case when $\mathcal{L}$ is the generator of a suitable Lévy process.Item Open Access Boundary Value Problems for the Laplace Equation on Convex Domains with Analytic Boundary(2018-04-07) Rockstroh , ParousiaIn this thesis we study boundary value problems for the Laplace equation on do mains with smooth boundary. Central to our analysis is a relation, known as the global relation, that couples the boundary data for a given BVP. Previously, the global re lation has primarily been applied to elliptic PDEs defined on polygonal domains. In this thesis we extend the use of the global relation to domains with smooth boundary. This is done by introducing a new transform, denoted by F_p, that is an analogue of the Fourier transform on smooth convex curves. We show that the F_p-transform is a bounded and invertible integral operator. Following this, we show that the F_p-transform naturally arises in the global relation for the Laplace equation on domains with smooth boundary. Using properties of the F_p-transform, we show that the global relation defines a continuously invertible map between the Dirichlet and Neumann data for a given BVP for the Laplace equation. Following this, we construct a numerical method that uses the global relation to find the Neumann data, given the Dirichlet data, for a given BVP for the Laplace equation on a domain with smooth boundary.Item Open Access Geometry of sub-Riemannian diffusion processes(2018-05-19) Habermann, Karen; Habermann, Karen [0000-0002-3533-900X]Sub-Riemannian geometry is the natural setting for studying dynamical systems, as noise often has a lower dimension than the dynamics it enters. This makes sub-Riemannian geometry an important field of study. In this thesis, we analysis some of the aspects of sub-Riemannian diffusion processes on manifolds. We first focus on studying the small-time asymptotics of sub-Riemannian diffusion bridges. After giving an overview of recent work by Bailleul, Mesnager and Norris on small-time fluctuations for the bridge of a sub-Riemannian diffusion, we show, by providing a specific example, that, unlike in the Riemannian case, small-time fluctuations for sub-Riemannian diffusion bridges can exhibit exotic behaviours, that is, qualitatively different behaviours compared to Brownian bridges. We further extend the analysis by Bailleul, Mesnager and Norris of small-time fluctuations for sub-Riemannian diffusion bridges, which assumes the initial and final positions to lie outside the sub-Riemannian cut locus, to the diagonal and describe the asymptotics of sub-Riemannian diffusion loops. We show that, in a suitable chart and after a suitable rescaling, the small-time diffusion loop measures have a non-degenerate limit, which we identify explicitly in terms of a certain local limit operator. Our analysis also allows us to determine the loop asymptotics under the scaling used to obtain a small-time Gaussian limit for the sub-Riemannian diffusion bridge measures by Bailleul, Mesnager and Norris. In general, these asymptotics are now degenerate and need no longer be Gaussian. We close by reporting on work in progress which aims to understand the behaviour of Brownian motion conditioned to have vanishing $N$th truncated signature in the limit as $N$ tends to infinity. So far, it has led to an analytic proof of the stand-alone result that a Brownian bridge in $\mathbb{R}^d$ from $0$ to $0$ in time $1$ is more likely to stay inside a box centred at the origin than any other Brownian bridge in time $1$.Item Open Access Scaling limit of critical systems in random geometry(2017-11-25) Powell, Ellen GraceThis thesis focusses on the properties of, and relationships between, several fundamental objects arising from critical physical models. In particular, we consider Schramm--Loewner evolutions, the Gaussian free field, Liouville quantum gravity and the Brownian continuum random tree. We begin by considering branching diffusions in a bounded domain $D\subset$ $R^{d}$, in which particles are killed upon hitting the boundary $\partial D$. It is known that such a system displays a phase transition in the branching rate: if it exceeds a critical value, the population will no longer become extinct almost surely. We prove that at criticality, under mild assumptions on the branching mechanism and diffusion, the genealogical tree associated with the process will converge to the Brownian CRT. Next, we move on to study Gaussian multiplicative chaos. This is the rigorous framework that allows one to make sense of random measures built from rough Gaussian fields, and again there is a parameter associated with the model in which a phase transition occurs. We prove a uniqueness and convergence result for approximations to these measures at criticality. From this point onwards we restrict our attention to two-dimensional models. First, we give an alternative, ``non-Gaussian" construction of Liouville quantum gravity (a special case of Gaussian multiplicative chaos associated with the 2-dimensional Gaussian free field), that is motivated by the theory of multiplicative cascades. We prove that the Liouville (GMC) measures associated with the Gaussian free field can be approximated using certain sequences of ``local sets" of the field. This is a particularly natural construction as it is both local and conformally invariant. It includes the case of nested CLE$_{4}$, when it is coupled with the GFF as its set of ``level lines". Finally, we consider this level line coupling more closely, now when it is between SLE$_{4}$ and the GFF. We prove that level lines can be defined for the GFF with a wide range of boundary conditions, and are given by SLE$_{4}$-type curves. As a consequence, we extend the definition of SLE$_{4}(\rho)$ to the case of a continuum of force points.Item Open Access Well-posedness and scattering of the Chern-Simons-Schrödinger system(2017-10-01) Lim, Zhuo MinThe subject of the present thesis is the Chern-Simons-Schrödinger system, which is a gauge-covariant Schrödinger system in two spatial dimensions with a long-range electromagnetic field. The present thesis studies two aspects of the system: that of well-posedness and that of the long-time behaviour. The first main result of the thesis concerns the large-data well-posedness of the initial-value problem for the Chern-Simons-Schrödinger system. We impose the Coulomb gauge to remove the gauge-invariance, in order to obtain a well-defined initial-value problem. We prove that, in the Coulomb gauge, the Chern-Simons-Schrödinger system is locally well-posed in the Sobolev spaces $H^s$ for $s\ge 1$, and that the solution map satisfies a weak Lipschitz continuity estimate. The main technical difficulty is the presence of a derivative nonlinearity, which rules out the naive iteration scheme for proving well-posedness. The key idea is to retain the non-perturbative part of the derivative nonlinearity in the principal operator, and to exploit the dispersive properties of the resulting paradifferential-type principal operator, in particular frequency-localised Strichartz estimates, using adaptations of the $U^p$ and $V^p$ spaces introduced by Koch and Tataru in other contexts. The other main result of the thesis characterises the large-time behaviour in the case where the interaction potential is the defocusing cubic term. We prove that the solution to the Chern-Simons-Schrödinger system in the Coulomb gauge, starting from a localised finite-energy initial datum, will scatter to a free Schrödinger wave at large times. The two crucial ingredients here are the discovery of a new conserved quantity, that of a pseudo-conformal energy, and the cubic null structure discovered by Oh and Pusateri, which reveals a subtle cancellation in the long-range electromagnetic effects. By exploiting pseudo-conformal symmetry, we also prove the existence of wave operators for the Chern-Simons-Schrödinger system in the Coulomb gauge: given a localised finite-energy final state, there exists a unique solution which scatters to that prescribed state.Item Open Access Deformation theory of Cayley submanifolds(2017-05-04) Moore, KimberleyCayley submanifolds are naturally arising volume minimising submanifolds of $Spin(7)$- manifolds. In the special case that the ambient manifold is a four-dimensional Calabi--Yau manifold, a Cayley submanifold might be a complex surface, a special Lagrangian submanifold or neither. In this thesis, we study the deformation theory of Cayley submanifolds from two different perspectives.Item Open Access Isospectral algorithms, Toeplitz matrices and orthogonal polynomials(2017-04-01) Webb, Marcus David; Webb, Marcus David [0000-0002-9440-5361]An isospectral algorithm is one which manipulates a matrix without changing its spectrum. In this thesis we study three interrelated examples of isospectral algorithms, all pertaining to Toeplitz matrices in some fashion, and one directly involving orthogonal polynomials. The first set of algorithms we study come from discretising a continuous isospectral flow designed to converge to a symmetric Toeplitz matrix with prescribed eigenvalues. We analyse constrained, isospectral gradient flow approaches and an isospectral flow studied by Chu in 1993. The second set of algorithms compute the spectral measure of a Jacobi operator, which is the weight function for the associated orthogonal polynomials and can include a singular part. The connection coefficients matrix, which converts between different bases of orthogonal polynomials, is shown to be a useful new tool in the spectral theory of Jacobi operators. When the Jacobi operator is a finite rank perturbation of Toeplitz, here called pert-Toeplitz, the connection coefficients matrix produces an explicit, computable formula for the spectral measure. Generalisation to trace class perturbations is also considered. The third algorithm is the infinite dimensional QL algorithm. In contrast to the finite dimensional case in which the QL and QR algorithms are equivalent, we find that the QL factorisations do not always exist, but that it is possible, at least in the case of pert-Toeplitz Jacobi operators, to implement shifts to generate rapid convergence of the top left entry to an eigenvalue. A fascinating novelty here is that the infinite dimensional matrices are computed in their entirety and stored in tailor made data structures. Lastly, the connection coefficients matrix and the orthogonal transformations computed in the QL iterations can be combined to transform these pert-Toeplitz Jacobi operators isospectrally to a canonical form. This allows us to implement a functional calculus for pert-Toeplitz Jacobi operators.