In this paper, a mixed boundary value problem for the Laplace-Beltrami operator is considered for spherical domains in $R^3$, i.e. for domains on the unit sphere. These domains are parametrized by spherical coordinates (\varphi, \theta), such that functions on the unit sphere are considered as functions in these coordinates. Careful investigation leads to the introduction of a proper finite element space corresponding to an isotropic triangulation of the underlying domain on the unit sphere. Error estimates are proven for a Clément-type interpolation operator, where appropriate, weighted norms are used. The estimates are applied to the deduction of a reliable and efficient residual error estimator for the Laplace-Beltrami operator.

Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. The simplest local error estimator from the implementation point of view is the so-called hierarchical error estimator. The reliability proof is usually based on two prerequisites: the saturation assumption and the strengthened Cauchy-Schwarz inequality. The proofs of these facts are extended in the present work for the case of the singularly perturbed reaction-diffusion equation and of the meshes with anisotropic elements. It is shown that the constants in the corresponding estimates do neither depend on the aspect ratio of the elements, nor on the perturbation parameters. Utilizing the above arguments the concluding reliability proof is provided as well as the efficiency proof of the estimator, both independent of the aspect ratio and perturbation parameters.

The Laplace-Beltrami operator corresponds to the Laplace operator on curved surfaces. In this paper, we consider an eigenvalue problem for the Laplace-Beltrami operator on subdomains of the unit sphere in $\R^3$. We develop a residual a posteriori error estimator for the eigenpairs and derive a reliable estimate for the eigenvalues. A global parametrization of the spherical domains and a carefully chosen finite element discretization allows us to use an approach similar to the one for the two-dimensional case. In order to assure results in the quality of those for plane domains, weighted norms and an adapted Clément-type interpolation operator have to be introduced.

In this thesis, we hope to dispel some confusion regarding the so-called post hoc power, i.e. power computed making the assumption that the estimated sample effect is equal to the population effect size. In previous research, it has been shown that post hoc power is a function of the p-value, making it redundant as a tool of analysis. We go further, arguing for it to never be reported, since it is a source of confusion and potentially harmful incentives. We also conduct a Monte Carlo simulation to illustrate our points of view. Previous research is confirmed by the results of this study.

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.

La séparation de sources avec un seul capteur est un problème très récent, qui attire de plus en plus d'attention dans le monde scientifique. Cependant, il est loin d'être résolu et, même plus, il ne peut pas être résolu en toute généralité. La difficulté principale est que, ce problème étant extrêmement sous déterminé, il faut disposer de fortes connaissances sur les sources pour pouvoir les séparer. Pour une grande partie des méthodes de séparation, ces connaissances sont représentées par des modèles statistiques des sources, notamment par des Modèles de Mélange de Gaussiennes (MMG), qui sont appris auparavant à partir d'exemples. L'objet de cette thèse est d'étudier les méthodes de séparation basées sur des modèles statistiques en général, puis de les appliquer à un problème concret, tel que la séparation de la voix par rapport à la musique dans des enregistrements monophoniques de chansons. Apporter des solutions à ce problème, qui est assez difficile et peu étudié pour l'instant, peut être très utile pour faciliter l'analyse du contenu des chansons, par exemple dans le contexte de l'indexation audio. Les méthodes de séparation existantes donnent de bonnes performances à condition que les caractéristiques des modèles statistiques utilisés soient proches de celles des sources à séparer. Malheureusement, il n'est pas toujours possible de construire et d'utiliser en pratique de tels modèles, à cause de l'insuffisance des exemples d'apprentissage représentatifs et des ressources calculatoires. Pour remédier à ce problème, il est proposé dans cette thèse d'adapter a posteriori les modèles aux sources à séparer. Ainsi, un formalisme général d'adaptation est développé. En s'inspirant de techniques similaires utilisées en reconnaissance de la parole, ce formalisme est introduit sous la forme d'un critère d'adaptation Maximum A Posteriori (MAP). De plus, il est montré comment optimiser ce critère à l'aide de l'algorithme EM à différents niveaux de généralité. Ce formalisme d'adaptation est ensuite appliqué dans certaines formes particulières pour la séparation voix / musique. Les résultats obtenus montrent que pour cette tâche, l'utilisation des modèles adaptés permet d'augmenter significativement (au moins de 5 dB) les performances de séparation par rapport aux modèles non adaptés. Par ailleurs, il est observé que la séparation de la voix chantée facilite l'estimation de sa fréquence fondamentale (pitch), et que l'adaptation des modèles ne fait qu'améliorer ce résultat.

Dans cette thèse, nous analysons une méthode de Galerkine discontinue (GD) et deux estimateurs d'erreur a posteriori pour l'équation d'advection-diffusion-réaction linéaire et stationnaire avec diffusion hétérogène. La méthode GD considérée, la méthode SWIP, utilise des moyennes pondérées dont les poids dépendent de la diffusion. L'analyse a priori montre que la convergence est optimale en le pas du maillage et robuste par rapport aux hétérogénéités de la diffusion, ce qui est confirmé par les tests numériques. Les deux estimateurs d'erreur a posteriori sont obtenus par une analyse par résidus et contrôlent la (semi-)norme d'énergie de l'erreur. L'analyse d'efficacité locale montre que presque tous les estimateurs sont indépendants des hétérogénéités. Le deuxième estimateur d'erreur est plus précis que le premier, mais son coût de calcul est légèrement plus élevé. Cet estimateur est basé sur la construction d'un flux H(div)-conforme dans l'espace de Raviart-Thomas-Nédéléc.

Simulating the human heart is a challenging problem, with simulations being very time consuming, to the extent that some can take days to compute even on high performance computing resources. There is considerable interest in computational optimisation techniques, with a view to making whole-heart simulations tractable. Reliability of heart model simulations is also of great concern, particularly considering clinical applications. Simulation software should be easily testable and maintainable, which is often not the case with extensively hand-optimised software. It is thus crucial to automate and verify any optimisations. CellML is an XML language designed for describing biological cell models from a mathematical modeller’s perspective, and is being developed at the University of Auckland. It gives us an abstract format for such models, and from a computer science perspective looks like a domain specific programming language. We are investigating the gains available from exploiting this viewpoint. We describe various static checks for CellML models, notably checking the dimensional consistency of mathematics, and investigate the possibilities of provably correct optimisations. In particular, we demonstrate that partial evaluation is a promising technique for this purpose, and that it combines well with a lookup table technique, commonly used in cardiac modelling, which we have automated. We have developed a formal operational semantics for CellML, which enables us to mathematically prove the partial evaluation of CellML correct, in that optimisation of models will not change the results of simulations. The use of lookup tables involves an approximation, thus introduces some error; we have analysed this using a posteriori techniques and shown how it may be managed. While the techniques could be applied more widely to biological models in general, this work focuses on cardiac models as an application area. We present experimental results demonstrating the effectiveness of our optimisations on a representative sample of cardiac cell models, in a variety of settings.

The presented work deals with the discontinuous Galerkin method with the anisotropic mesh adaptation for stationary convection-diffusion problems. Basic definitions are included in an introduction where we also present the used method. The following parts describe various methods for evaluating a Riemann metric, which is necessary for anisotropic mesh adaptation. The most important part of work follows - numerical experiments carried out with ADGFEM and ANGENER software packages. In these experiments, we compare different approaches for the definition of Riemann metrics and compare their efficiency. The main output of this thesis are subroutines for evaluation of the Riemann metric including its source code.

The thesis presents a comprehensive study of a posteriori error estimation in the adaptive solution to some classical elliptic partial differential equations. Several new error estimators are proposed for diffusion problems with discontinuous coefficients and for convection-reaction-diffusion problems with dominated convection/reaction. The robustness of the new estimators is justified theoretically. Extensive numerical results demonstrate the robustness of the new estimators for challenging problems and indicate that, compared to the well-known residual-type estimators, the new estimators are much more accurate.