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We study heat transfer in one dimension with and without convection, also called advection-diffusion. This is done using the Finite Element Method (FEM) to discretise the mathematical model, i.e. the heat equation. The results are compared to analytic Fourier series solutions. Our main result is that the FEM could be used to better model the heat transfer which allow for more accurate models than today's use of steady state models.

Abstract We are interested in studying Gaussian Markov random fields as correlation priors for Bayesian inversion. We construct the correlation priors to be discretisation-invariant, which means, loosely speaking, that the discrete priors converge to continuous priors at the discretisation limit. We construct the priors with stochastic partial differential equations, which guarantees computational efficiency via sparse matrix approximations. The stationary correlation priors have a clear statistical interpretation through the autocorrelation function. We also consider how to make structural model of an unknown object with anisotropic and inhomogeneous Gaussian Markov random fields. Finally we consider these fields on unstructured meshes, which are needed on complex domains. The publications in this thesis contain fundamental mathematical and computational results of correlation priors. We have considered one application in this thesis, the electrical impedance tomography. These fundamental results and application provide a platform for engineers and researchers to use correlation priors in other inverse problem applications.

L’enjeu de la simulation de la dynamique atmosphérique et océanographique a pris ces dernières années une importance accrue avec la question du réchauffement climatique. Le modèle à simuler est complexe. Il combine les équations de la mécanique des fluides avec celles de la thermodynamique. Au 19ème siècle, le mathématicien Adhémar Barré de Saint-Venant formule un système d’équations aux dérivées partielles décrivant les mouvements d’un fluide soumis à la gravité et de faible épaisseur. Il s’agit des équations Shallow Water. L’objectif de cette thèse est de développer et d’analyser un algorithme de résolution des équations Shallow Water sur une sphère en rotation. Dans un premier temps, j’étudie différents aspects mathématiques des opérateurs aux différences finis utilisés par la suite en géométrie sphérique. Les schémas aux différences obtenus sont utilisés pour résoudre l’équation de transport, l’équation des ondes et l’équation de Burgers. Les propriétés de stabilité précision et conservation sont analysées. Dans un second temps, la grille Cubed-Sphere est introduite et analysée. La structure de ce maillage est analogue à celle d’un cube. L'interprétation de la Cubed-Sphere à l’aide de grands cercles permet de construire des opérateurs sphériques discrets gradient, divergence et vorticité d'ordre au moins égal à 3 (en pratique d'ordre 4). La troisième partie de la thèse est dédiée à différents tests pour le système d’équations Shallow Water ainsi que pour l’équation d’advection. Les résultats démontrent une précision proche de celle obtenue par les algorithmes conservatifs d'ordre 4 les plus récents / The problem to obtain accurate simulations of the atmospheric and oceanic equations has become essential in recent years for a proper understanding of the climate change. The full mathematical model to simulate is rather complex. It consists of the coupling of several equations involving fluid dynamics and thermodynamics. In the 19th century, Adhémar Barré de Saint-Venant first formulated the equations describing the dynamic of a fluid subject to gravity and bottom topography. This system is Shallow Water equations. The goal of this thesis is to develop and analyze a numerical scheme to solve the shallow water equation on a rotating sphere. First, a mathematical analsysis of finite difference operators that will be used on the sphere is presented. These schemes are then used to solve various equations in a spehreical setting, in particular the advection equation, the wave equation and the Burgers equation. Stability, accuracy and conservation properties are studied. In a second part, I consider in detail the Cubed-Sphere grid. This particular spherical grid has the mesh topology of a cube. Another interpretation makes use of great circles, this allows to obtain spherical discret operators gradient, divergence and curl of a preved third order. These operators are numercially of 4th order. Numerial results are show in particular for the SW equations an acurracy similar to the one of conservative schemes of 4th order published recently

This dissertation focuses on the further development of creating accurate numerical models of complex dynamical systems using the holistic discretisation technique [Roberts, Appl. Num. Model., 37:371-396, 2001]. I extend the application from second to fourth order systems and from only one spatial dimension in all previous work to two dimensions (2D). We see that the holistic technique provides useful and accurate numerical discretisations on coarse grids. We explore techniques to model the evolution of spatial patterns governed by pdes such as the Kuramoto-Sivashinsky equation and the real-valued Ginzburg-Landau equation. We aim towards the simulation of fluid flow and convection in three spatial dimensions. I show that significant steps have been taken in this dissertation towards achieving this aim. Holistic discretisation is based upon centre manifold theory [Carr, Applications of centre manifold theory, 1981] so we are assured that the numerical discretisation accurately models the dynamical system and may be constructed systematically. To apply centre manifold theory the domain is divided into elements and using a homotopy in the coupling parameter, subgrid scale fields are constructed consisting of actual solutions of the governing partial differential equation(pde). These subgrid scale fields interact through the introduction of artificial internal boundary conditions. View the centre manifold (macroscale) as the union of all states of the collection of subgrid fields (microscale) over the physical domain. Here we explore how to extend holistic discretisation to the fourth order Kuramoto-Sivashinsky pde. I show that the holistic models give impressive accuracy for reproducing the steady states and time dependent phenomena of the Kuramoto-Sivashinsky equation on coarse grids. The holistic method based on local dynamics compares favourably to the global methods of approximate inertial manifolds. The excellent performance of the holistic models shown here is strong evidence in support of the holistic discretisation technique. For shear dispersion in a 2D channel a one-dimensional numerical approximation is generated directly from the two-dimensional advection-diffusion dynamics. We find that a low order holistic model contains the shear dispersion term of the Taylor model [Taylor, IMA J. Appl. Math., 225:473-477, 1954]. This new approach does not require the assumption of large x scales, formerly absolutely crucial in deriving the Taylor model. I develop holistic discretisation for two spatial dimensions by applying the technique to the real-valued Ginzburg-Landau equation as a representative example of second order pdes. The techniques will apply quite generally to second order reaction-diffusion equations in 2D. This is the first study implementing holistic discretisation in more than one spatial dimension. The previous applications of holistic discretisation have developed algebraic forms of the subgrid field and its evolution. I develop an algorithm for numerical construction of the subgrid field and its evolution for 1D and 2D pdes and explore various alternatives. This new development greatly extends the class of problems that may be discretised by the holistic technique. This is a vital step for the application of the holistic technique to higher spatial dimensions and towards discretising the Navier-Stokes equations.

Control of the transition of laminar flow to turbulence would result in lower drag and reduced energy consumption in many engineering applications. A spectral state-space model of linearised plane Poiseuille flow with wall transpiration ac¬tuation and wall shear measurements is developed from the Navier-Stokes and continuity equations, and optimal controllers are synthesized and assessed in sim¬ulations of the flow. The polynomial-form collocation model with control by rate of change of wall-normal velocity is shown to be consistent with previous interpo¬lating models with control by wall-normal velocity. Previous methods of applying the Dirichlet and Neumann boundary conditions to Chebyshev series are shown to be not strictly valid. A partly novel method provides the best numerical behaviour after preconditioning. Two test cases representing the earliest stages of the transition are consid¬ered, and linear quadratic regulators (LQR) and estimators (LQE) are synthesized. Finer discretisation is required for convergence of estimators. A novel estimator covariance weighting improves estimator transient convergence. Initial conditions which generate the highest subsequent transient energy are calculated. Non-linear open- and closed-loop simulations, using an independently derived finite-volume Navier-Stokes solver modified to work in terms of perturbations, agree with linear simulations for small perturbations. Although the transpiration considered is zero net mass flow, large amounts of fluid are required locally. At larger perturbations the flow saturates. State feedback controllers continue to stabilise the flow, but estimators may overshoot and occasionally output feedback destabilises the flow. Actuation by simultaneous wall-normal and tangential transpiration is derived. There are indications that control via tangential actuation produces lower highest transient energy, although requiring larger control effort. State feedback controllers are also synthesized which minimise upper bounds on the highest transient energy and control effort. The performance of these controllers is similar to that of the optimal controllers.

Control of the transition of laminar flow to turbulence would result in lower drag and reduced energy consumption in many engineering applications. A spectral state-space model of linearised plane Poiseuille flow with wall transpiration ac¬tuation and wall shear measurements is developed from the Navier-Stokes and continuity equations, and optimal controllers are synthesized and assessed in sim¬ulations of the flow. The polynomial-form collocation model with control by rate of change of wall-normal velocity is shown to be consistent with previous interpo¬lating models with control by wall-normal velocity. Previous methods of applying the Dirichlet and Neumann boundary conditions to Chebyshev series are shown to be not strictly valid. A partly novel method provides the best numerical behaviour after preconditioning. Two test cases representing the earliest stages of the transition are consid¬ered, and linear quadratic regulators (LQR) and estimators (LQE) are synthesized. Finer discretisation is required for convergence of estimators. A novel estimator covariance weighting improves estimator transient convergence. Initial conditions which generate the highest subsequent transient energy are calculated. Non-linear open- and closed-loop simulations, using an independently derived finite-volume Navier-Stokes solver modified to work in terms of perturbations, agree with linear simulations for small perturbations. Although the transpiration considered is zero net mass flow, large amounts of fluid are required locally. At larger perturbations the flow saturates. State feedback controllers continue to stabilise the flow, but estimators may overshoot and occasionally output feedback destabilises the flow. Actuation by simultaneous wall-normal and tangential transpiration is derived. There are indications that control via tangential actuation produces lower highest transient energy, although requiring larger control effort. State feedback controllers are also synthesized which minimise upper bounds on the highest transient energy and control effort. The performance of these controllers is similar to that of the optimal controllers.

This thesis contributes to the field of causal inference, where the main interest is to estimate the effect of a treatment on some outcome. At its core, causal inference is an exercise in controlling for imbalance (differences) in covariate distributions between the treated and the controls, as such imbalances otherwise can bias estimates of causal effects. Imbalance on observed covariates can be handled through matching, where treated and controls with similar covariate distributions are extracted from a data set and then used to estimate the effect of a treatment. The first paper of this thesis describes and investigates a matching design, where a data-driven algorithm is used to discretise a covariate before matching. The paper also gives sufficient conditions for if, and how, a covariate can be discretised without introducing bias. Balance is needed for unobserved covariates too, but is more difficult to achieve and verify. Unobserved covariates are sometimes replaced with correlated counterparts, usually referred to as proxy variables. However, just replacing an unobserved covariate with a correlated one does not guarantee an elimination of, or even reduction of, bias. In the second paper we formalise proxy variables in a causal inference framework and give sufficient conditions for when they lead to nonparametric identification of causal effects. The third and fourth papers both concern estimating the effect an enhanced cooperation between the Swedish Social Insurance Agency and the Public Employment Service has on reducing sick leave. The third paper is a study protocol, where the matching design used to estimate this effect is described. The matching was then also carried out in the study protocol, before the outcome for the treated was available, ensuring that the matching design was not influenced by any estimated causal effects. The third paper also presents a potential proxy variable for unobserved covariates, that is used as part of the matching. The fourth paper then carries out the analysis described in the third paper, and uses an instrumental variable approach to test for unobserved confounding not captured by the supposed proxy variable.

In der Finanzmathematik hat der Besitzer einer amerikanische Option das Recht aber nicht die Pflicht, eine Aktie innerhalb eines bestimmten Zeitraums, für einen bestimmten Preis zu kaufen oder zu verkaufen. Die Bewertung einer amerikanische Option wird als so genanntes optimale stopping Problem formuliert. Erfolgt die Modellierung des Aktienkurses durch eine geometrische Brownsche Bewegung, wird der Wert einer amerikanischen Option durch ein deterministisches freies Randwertproblem (FRWP), oder einer äquivalenten Variationsungleichung (VU) auf ganz R in gewichteten Sobolev Räumen gegeben. Um Standardmethoden der Numerischen Mathematik anzuwenden, wird das unbeschränkte Gebiet zu einem beschränkten Gebiet abgeschnitten. Mit Hilfe der Fourier-Transformation wird eine Integraldarstellung der Lösung die den freien Rand explizit beinhaltet, hergeleitet. Mittels dieser Integraldarstellung werden Abschneidefehlerschranken bewiesen. Danach werden gewichtete Poincare Ungleichungen mit expliziten Konstanten bewiesen. Der Abschneidefehler und die gewichtete Poincare Ungleichung ermöglichen, einen zuverlässigen a posteriori Fehlerschätzer zwischen der exakten Lösung der VU und der semidiskreten Lösung des penalisierten Problems auf R herzuleiten. Eine hinreichend glatte Lösung der VU garantiert die Konvergenz der Lösung des penaltisierten Problems zur Lösung der VU. Ein a priori Fehlerschätzer für den Fehler zwischen der exakten Lösung der VU und der semidiskreten Lösung des penaltisierten Problems beendet die numerische Analysis. Die eingeführten aposteriori Fehlerschätzer motivieren einen Algorithmus für adaptive Netzverfeinerung. Numerische Experimente zeigen die verbesserte Konvergenz des adaptiven Verfahrens gegenüber der uniformen Verfeinerung. Der zuverlässige a posteriori Fehlerschätzer ermöglicht es, den Abschneidepunkt so zu wählen, dass der Gesamtfehler (Diskretisierungsfehler plus Abschneidefehler) kleiner als eine gegebenen Toleranz ist. / Among the central concerns in mathematical finance is the evaluation of American options. An American option gives the holder the right but not the obligation to buy or sell a certain financial asset within a certain time-frame, for a certain strike price. The valuation of American options is formulated as an optimal stopping problem. If the stock price is modelled by a geometric Brownian motion, the value of an American option is given by a deterministic parabolic free boundary value problem (FBVP) or equivalently a non-symmetric variational inequality (VI) on weighted Sobolev spaces on R. To apply standard numerical methods, the unbounded domain R is truncated to a bounded one. Applying the Fourier transform to the FBVP yields an integral representation of the solution including the free boundary explicitely. This integral representation allows to prove explicit truncation errors. Since the VI is formulated within the framework of weighted Sobolev spaces, we establish a weighted Poincare inequality with explicit determined constants. The truncation error estimate and the weighted Poncare inequality enable a reliable a posteriori error estimate between the exact solution of the VI and the semi-discrete solution of the penalised problem on R. A sufficient regular solution provides the convergence of the solution of the penalised problem to the solution of the VI. An a priori error estimate for the error between the exact solution of the VI and the semi-discrete solution of the penalised problem concludes the numerical analysis. The established a posteriori error estimates motivates an algorithm for adaptive mesh refinement. Numerical experiments show the improved convergence of the adaptive algorithm compared to uniform mesh refinement. The reliable a posteriori error estimate including explicit truncation errors allows to determine a truncation point such that the total error (discretisation and truncation error) is below a given error tolerance.

Die Dissertation behandelt methodische und algorithmische Aspekte der Spektralelementemethode zur räumlichen Diskretisierung partieller Differentialgleichungen. Die Weiterentwicklung einer symmetriebasierten Faktorisierung ermöglicht effiziente Operatoren für Tetraederelemente. Auf Grundlage einer umfassenden Leistungsanalyse werden Engpässe in der Implementierung der Operatoren identifiziert und durch algorithmische Modifikationen der Methode eliminiert.