In this thesis we study three stochastic partial differential equations (SPDE) that arise as stochastic gradient flows via the fluctuation-dissipation principle. For the first equation we establish a finer regularity statement based on a generalized Taylor expansion which is inspired by the theory of rough paths. The second equation is the thin-film equation with thermal noise which is a singular SPDE. In order to circumvent the issue of dealing with possible renormalization, we discretize the gradient flow structure of the deterministic thin-film equation. Choosing a specific discretization of the metric tensor, we resdiscover a well-known discretization of the thin-film equation introduced by Grün and Rumpf that satisfies a discrete entropy estimate. By proving a stochastic entropy estimate in this discrete setting, we obtain positivity of the scheme in the case of no-slip boundary conditions. Moreover, we analyze the associated rate functional and perform numerical experiments which suggest that the scheme converges. The third equation is the massive $\varphi^4_2$-model on the torus which is also a singular SPDE. In the spirit of Bakry and Émery, we obtain a gradient bound on the Markov semigroup. The proof relies on an $L^2$-estimate for the linearization of the equation. Due to the required renormalization, we use a stopping time argument in order to ensure stochastic integrability of the random constant in the estimate. A postprocessing of this estimate yields an even sharper gradient bound. As a corollary, for large enough mass, we establish a local spectral gap inequality which by ergodicity yields a spectral gap inequality for the $\varphi^4_2$- measure.

We first provide a detailed background of a geometric projection methodology developed by Professor Roswitha Marz at Humboldt University in Berlin for showing uniqueness and existence of solutions for ordinary differential-algebraic equations (DAEs). Because of the geometric and operator-theoretic aspects of this particular method, it can be extended to the case of infinite-dimensional abstract DAEs. For example, partial differential equations (PDEs) are often formulated as abstract Cauchy or evolution problems which we label abstract ordinary differential equations or AODE. Using this abstract formulation, existence and uniqueness of the Cauchy problem has been studied. Similarly, we look at an AODE system with operator constraint equations to formulate an abstract differential-algebraic equation or ADAE problem. Existence and uniqueness of solutions is shown under certain conditions on the operators for both index-1 and index-2 abstract DAEs. These existence and uniqueness results are then applied to some index-1 DAEs in the area of thermodynamic modeling of a chemical vapor deposition reactor and to a structural dynamics problem. The application for the structural dynamics problem, in particular, provides a detailed construction of the model and development of the DAE framework. Existence and uniqueness are primarily demonstrated using a semigroup approach. Finally, an exploration of some issues which arise from discretizing the abstract DAE are discussed. / Ph. D.

Interval analysis is an essential tool in the construction of validated numerical solutions of Initial Value Problems (IVP) for Ordinary (ODE) and Partial (PDE) Differential Equations. A validated solution typically consists of guaranteed lower and upper bounds for the exact solution or set of exact solutions in the case of uncertain data, i.e. it is an interval function (enclosure) containing all solutions of the problem. IVP for ODE: The central point of discussion is the wrapping effect. A new concept of wrapping function is introduced and applied in studying this effect. It is proved that the wrapping function is the limit of the enclosures produced by any method of certain type (propagate and wrap type). Then, the wrapping effect can be quantified as the difference between the wrapping function and the optimal interval enclosure of the solution set (or some norm of it). The problems with no wrapping effect are characterized as problems for which the wrapping function equals the optimal interval enclosure. A sufficient condition for no wrapping effect is that there exist a linear transformation, preserving the intervals, which reduces the right-hand side of the system of ODE to a quasi-isotone function. This condition is also necessary for linear problems and "near" necessary in the general case. Hyperbolic PDE: The Initial Value Problem with periodic boundary conditions for the wave equation is considered. It is proved that under certain conditions the problem is an operator equation with an operator of monotone type. Using the established monotone properties, an interval (validated) method for numerical solution of the problem is proposed. The solution is obtained step by step in the time dimension as a Fourier series of the space variable and a polynomial of the time variable. The numerical implementation involves computations in Fourier and Taylor functoids. Propagation of discontinuo~swaves is a serious problem when a Fourier series is used (Gibbs phenomenon, etc.). We propose the combined use of periodic splines and Fourier series for representing discontinuous functions and a method for propagating discontinuous waves. The numerical implementation involves computations in a Fourier hyper functoid. / Mathematical Sciences / D. Phil. (Mathematics)

Using a Lie symmetry group generator and a generalized form of Manale's formula for solving second order ordinary di erential equations, we determine new symmetries for the one and two dimensional heat equations, leading to new solutions. As an application, we test a formula resulting from this approach on thin plate heat conduction. / Applied Mathematics / M.Sc. (Applied Mathematics)

In this thesis, we develop preconditioned iterative methods for the solution of matrix systems arising from PDE-constrained optimization problems. In order to do this, we exploit saddle point theory, as this is the form of the matrix systems we wish to solve. We utilize well-known results on saddle point systems to motivate preconditioners based on effective approximations of the (1,1)-block and Schur complement of the matrices involved. These preconditioners are used in conjunction with suitable iterative solvers, which include MINRES, non-standard Conjugate Gradients, GMRES and BiCG. The solvers we use are selected based on the particular problem and preconditioning strategy employed. We consider the numerical solution of a range of PDE-constrained optimization problems, namely the distributed control, Neumann boundary control and subdomain control of Poisson's equation, convection-diffusion control, Stokes and Navier-Stokes control, the optimal control of the heat equation, and the optimal control of reaction-diffusion problems arising in chemical processes. Each of these problems has a special structure which we make use of when developing our preconditioners, and specific techniques and approximations are required for each problem. In each case, we motivate and derive our preconditioners, obtain eigenvalue bounds for the preconditioners where relevant, and demonstrate the effectiveness of our strategies through numerical experiments. The goal throughout this work is for our iterative solvers to be feasible and reliable, but also robust with respect to the parameters involved in the problems we consider.

The first contribution of this thesis is a new regularity theorem for time harmonic Maxwell's equations with less than Lipschitz complex anisotropic coefficients. By using the L^p theory for elliptic equations, it is possible to prove H¹ and Hölder regularity results, provided that the coefficients are W^1,p for some p = 3. This improves previous regularity results, where the assumption W^1,∞ for the coefficients was believed to be optimal. The method can be easily extended to the case of bi-anisotropic materials, for which a separate approach turns out to be unnecessary. The second focus of this work is the boundary control of the Helmholtz and Maxwell equations to enforce local constraints inside the domain. More precisely, we look for suitable boundary conditions such that the corresponding solutions and their derivatives satisfy certain local non-zero constraints. Complex geometric optics solutions can be used to construct such illuminations, but are impractical for several reasons. We propose a constructive approach to this problem based on the use of multiple frequencies. The suitable boundary conditions are explicitly constructed and give the desired constraints, provided that a finite number of frequencies, given a priori, are chosen in a fixed range. This method is based on the holomorphicity of the solutions with respect to the frequency and on the regularity theory for the PDE under consideration. This theory finds applications to several hybrid imaging inverse problems, where the unknown coefficients have to be imaged from internal measurements. In order to perform the reconstruction, we often need to find suitable boundary conditions such that the corresponding solutions satisfy certain non-zero constraints, depending on the particular problem under consideration. The multiple frequency approach introduced in this thesis represents a valid alternative to the use of complex geometric optics solutions to construct such boundary conditions. Several examples are discussed.

The erosion and transport of fertile topsoil is a serious problem in the U.S., Australia, China and throughout Europe. It results in extensive environmental damage, reduces soil fertility and productivity, and causes significant environmental loss. It is as big a threat to the future sustainability of global populations as climate change, but receives far less attention. With both chemicals (fertilizers, pesticides, herbicides) and biological pathogens (bacteria, viruses) preferentially sorbing to silt and clay sized soil particles, estimating contaminant fluxes in eroded soil also requires predicting the transported soils particle size distribution. The Hairsine-Rose (HR) erosion model is considered in this thesis as it is one of the very few that is specifically designed to incorporate the effect of particle size distribution, and differentiates between non-cohesive previously eroded soil compared with cohesive un-eroded soil. This thesis develops a new extended erosion model that couples the HR approach with the one-dimensional St Venant equations, and an Exner bed evolution equation to allow for feedback effects from changes in the local bed slope on surface hydraulics and erosion rates to be included. The resulting system of 2I +3 (where I = number of particle size classes) nonlinear hyperbolic partial differential equations is then solved numerically using a Liska-Wendroff predictor corrector finite difference scheme. Approximate analytical solutions and series expansions are derived to overcome singularities in the numerical solutions arising from either boundary or initial conditions corresponding to a zero flow depth. Three separate practical applications of the extended HR model are then considered in this thesis, (i) flow through vegetative buffer strips, (ii) modelling discharge hysteresis loops and (iii) the growth of antidunes, transportational cyclic steps and travelling wave solutions. It is shown by comparison against published experimental flume data that predictions from the extended model are able to closely match measurements of deposited sediment distribution both upstream and within the vegetative buffer strip. The experiments were conducted with supercritical inflow to the flume which due to the increased drag from the vegetative strip, resulted in a hydraulic jump just upstream of the vegetation. As suspended sediment deposited at the jump, this resulted in the jump slowly migrating upstream. The numerical solutions were also able to predict the position and hydraulic jump and the flow depth throughout the flume, including within the vegetative strip, very well. In the second application, it is found that the extended HR model is the first one that can produce all known types of measured hysteresis loops in sediment discharge outlet data. Five main loop types occur (a) clockwise, (b) counter-clockwise, (c,d) figure 8 of both flow orientations and (e) single curve. It is clearly shown that complicated temporal rainfall patterns or bed geometry are not required to developed complicated hysteresis loops, but it is the spatial distribution of previously eroded sediment that remains for the start of a new erosion event, which primarily governs the form of the hysteresis loop. The role of the evolution of the sediment distribution in the deposited layer therefore controls loop shape and behavior. Erosion models that are based solely on suspended sediment are therefore unable to reproduce these hysteretic loops without a priori imposing a hysteretic relationship on the parameterisations of the erosion source terms. The rather surprising result that the loop shape is also dominated by the suspended concentration of the smallest particle size is shown and discussed. In the third application, a linear stability analysis shows that instabilities, antidunes, will grow and propagate upstream under supercritical flow conditions. Numerical simulations are carried out that confirm the stability analysis and show the development and movement of antidunes. For various initial parameter configurations a series of travelling antidunes, or transportational cyclic steps, separated by hydraulic jumps are shown to develop and evolve to a steady form and wave speed. Two different forms arise whereby (a) the deposited layer completely shields the underlying original cohesive soil so that the cohesive layer plays no role in the speed or shape of the wave profile or (b) the cohesive soil is exposed along the back of the wave such that both the non-cohesive and cohesive layers affect the wave profile. Under (a) the solutions are obtained up to an additive constant as the actual location of the boundary of the cohesive soil is not required, whereas for (b) this constant must be determined in order to find the location on the antidune from where the cohesive soil becomes accessible. For single size class soils the leading order travelling wave equations are fairly straightforward to obtain for both cases (a) and (b). However for multi-size class soils, this becomes much more demanding as up to 2I + 3 parameters must be found iteratively to define the solution as each size class has its own wave profile in suspension and in the antidune.

The present thesis is on the error estimates of different energy based quasicontinuum (QC) methods, which are a class of computational methods for the coupling of atomistic and continuum models for micro- or nano-scale materials. The thesis consists of two parts. The first part considers the a priori error estimates of three energy based QC methods. The second part deals with the a posteriori error estimates of a specific energy based QC method which was recently developed. In the first part, we develop a unified framework for the a priori error estimates and present a new and simpler proof based on negative-norm estimates, which essentially extends previous results. In the second part, we establish the a posteriori error estimates for the newly developed energy based QC method for an energy norm and for the total energy. The analysis is based on a posteriori residual and stability estimates. Adaptive mesh refinement algorithms based on these error estimators are formulated. In both parts, numerical experiments are presented to illustrate the results of our analysis and indicate the optimal convergence rates. The thesis is accompanied by a thorough introduction to the development of the QC methods and its numerical analysis, as well as an outlook of the future work in the conclusion.

A major problem in the quantitative analysis of AC voltammetric data has been the variance in results between laboratories, often resulting from a reliance on "heuristic" methods of parameter estimation that are strongly dependent on the choices of the operator. In this thesis, an automatic method for parameter estimation will be tested in the context of experiments involving electron-transfer processes in solution-phase. It will be shown that this automatic method produces parameter estimates consistent with those from other methods and the literature in the case of the ferri-/ferrocyanide couple, and is able to explain inconsistency in published values of the rate parameter for the ferrocene/ferrocenium couple. When a coupled homogeneous reaction is considered in a theoretical study, parameter recovery is achieved with a higher degree of accuracy when simulated data resulting from a high frequency AC voltammetry waveform are used. When surface-confined reactions are considered, heterogeneity in the rate constant and formal potential make parameter estimation more challenging. In the final study, a method for incorporating these "dispersion" effects into voltammetric simulations is presented, and for the first time, a quantitive theoretical study of the impact of dispersion on measured current is undertaken.

Chemotaxis, the phenomenon through which cells respond to external chemical signals, is one of the most important and universally observable in nature. It has been the object of considerable modelling effort in the last decades. The models for chemotaxis available in the literature cannot reconcile the dynamics of external chemical signals and the intracellular signalling pathways leading to the response of the cells. The reason is that models used for cells do not contain the distinction between the extracellular and intracellular domains. The work presented in this dissertation intends to resolve this issue. We set up a numerical hybrid simulation framework containing such description and enabling the coupling of models for phenomena occurring at extracellular and intracellular levels. Mathematically, this is achieved by the use of the fictitious domain method for finite elements, allowing the simulation of partial differential equations on evolving domains. In order to make the modelling of the membrane binding of chemical signals possible, we derive a suitable fictitious domain method for Robin boundary elliptic problems. We also display ways to minimise the computational cost of such simulation by deriving a suitable preconditioner for the linear systems resulting from the Robin fictitious domain method, as well as an efficient algorithm to compute fictitious domain specific linear operators. Lastly, we discuss the use of a simpler cell model from the literature and match it with our own model. Our numerical experiments show the relevance of the matching, as well as the stability and accuracy of the numerical scheme presented in the thesis.