Odorous sensations are based on trigeminal and olfactory perceptions. Both trigeminal and olfactory stimuli generate overlapping as well as distinctive activations in the olfactory cortex including the piriform cortex. Orbitofrontal cortex (OFC), an integrative center for all senses, is directly activated in the presence of olfactory stimulations. In contrast, the thalamus, a very important midbrain structure, is not directly activated in the presence of odors, but rather acts as a relay for portions of olfactory information between primary olfactory cortex and higher-order processing centers. The aims of the study were (1) to examine the number of streamlines between the piriform cortex and the OFC and also between the piriform cortex and the thalamus and (2) to explore potential correlations between these streamlines and trigeminal and olfactory chemosensory perceptions. Thirty-eight healthy subjects were recruited for the study and underwent diffusion MRI using a 3T MRI scanner with 67 diffusion directions. ROIs were adapted from two studies looking into olfaction in terms of functional and structural properties of the olfactory system. The “waytotal number” was used which corresponds to number of streamlines between two regions of interests. We found the number of streamlines between the piriform cortex and the thalamus to be higher in the left hemisphere, whereas the number of streamlines between the piriform cortex and the OFC were higher in the right hemisphere. We also found streamlines between the piriform cortex and the thalamus to be positively correlated with the intensity of irritating (trigeminal) odors. On the other hand, streamlines between the piriform cortex and the OFC were correlated with the threshold scores for these trigeminal odors. This is the first studying the correlations between streamlines and olfactory scores using tractography. Results suggest that different chemosensory stimuli are processed through different networks in the chemosensory system involving the thalamus.

People suffering from congenital anosmia show normal brain architecture although they do not have functional sense of smell. Some studies in this regard point to the changes in secondary olfactory cortex, orbitofrontal cortex (OFC), in terms of gray matter volume increase. However, diffusion tensor imaging has not been explored so far. We included 13 congenital anosmia subjects together with 15 controls and looked into various diffusion parameters like FA. Increased FA in bilateral OFC confirms the earlier studies reporting increased gray matter thickness. However, it is quite difficult to interpret FA in terms of gray matter volume. Increased FA has been seen with recovery after traumatic brain injury. Such changes in OFC point to the plastic nature of the brain.

Diese Arbeit behandelt die Asymptotik von Austritts- und Übergangszeiten für gewisse schwach zeitinhomogene Diffusionsprozesse. Darauf basierend wird ein probabilistischer Begriff der stochastischen Resonanz (SR) studiert. Techniken der großen Abweichungen spielen eine zentrale Rolle. Im ersten Teil der Arbeit (Kapitel 1-3) werden Resultate aus der Theorie der großen Abweichungen für zeithomogene Diffusionen rekapituliert. Es werden die klassischen Resultate von Freidlin und Wentzell und Erweiterungen dieser Theorie präsentiert, und es wird an das Kramers''sche Austrittszeitengesetz erinnert. Teil II befasst sich mit dem Phänomen der SR, d.h. mit Periodizitätseigenschaften von Diffusionen. In Kapitel 4 werden physikalische Maße zur Messung der Periodizität diskutiert. Deren Nachteile legen es nahe, einem alternativen, probabilistischen Ansatz zu folgen, der hier behandelt wird. Das 5. Kapitel dient der Herleitung eines gleichmäßigen Prinzips der großen Abweichungen für Diffusionen mit schwach zeitabhängigem, periodischem Drift. Die Gleichmäßigkeit des Prinzips ermöglicht die exakte Bestimmung exponentieller Übergangsraten in Kapitel 6, das die zentralen Ergebnisse des 2. Teils beinhaltet. Hierdurch wird die Maximierung gewisser Übergangswahrscheinlichkeiten ermöglicht, was zum in Kapitel 7 studierten Resonanzbegriff führt. Teil III der Arbeit setzt sich mit der Asymptotik von Austrittszeiten sogenannter selbststabilisierender Diffusionen auseinander. In Kapitel 8 wird der Zusammenhang zwischen interagierenden Teilchensystemen und selbststabilisierenden Diffusionen erläutert und die Existenz- und Eindeutigkeitsfrage behandelt. Das 9. Kapitel dient dem Studium der großen Abweichungen dieser Klasse von Diffusionen. In Kapitel 10 wird das Kramers''sche Austrittszeitengesetz auf selbststabilisierende Diffusionen übertragen, und in Kapitel 11 wird der Einfluß der selbststabilisierenden Komponente auf das Austrittszeitengesetz illustriert. / In this thesis, we study the asymptotic behavior of exit and transition times of certain weakly time inhomogeneous diffusion processes. Based on these asymptotics, a probabilistic notion of stochastic resonance (SR) is investigated. Large deviations techniques play the key role throughout this work. In the first part (Chapters 1-3) we recall the large deviations theory for time homogeneous diffusions. We present the classical results due to Freidlin and Wentzell and extensions thereof, and we remind of Kramers'' exit time law. Part II deals with the phenomenon of stochastic resonance. That is, we study periodicity properties of diffusion processes. In Chapter 4 we explain the paradigm of stochastic resonance and discuss physical notions of measuring periodicity of diffusions. Their drawbacks suggest to follow an alternative probabilistic approach, which is treated in this work. In Chapter 5 we derive a large deviations principle for diffusions subject to a weakly time dependent periodic drift term. The uniformity of the obtained large deviations bounds w.r.t. the system''s parameters plays a key role for the treatment of transition time asymptotics in Chapter 6, which contains the main result of the second part. The exact exponential transition rates obtained here allow for maximizing transition probabilities, which finally leads to the announced probabilistic notion of resonance studied in Chapter 7. In the third part we investigate the exit time asymptotics of a certain class of so-called self-stabilizing diffusions. In Chapter 8 we explain the connection between interacting particle systems and self-stabilizing diffusions, and we address the question of existence. The following Chapter 9 is devoted to the study of the large deviations behavior of these diffusions. In Chapter 10 Kramers'' exit law is carried over to our class of self-stabilizing diffusions. Finally, the influence of self-stabilization is illustrated in Chapter 11.

Das globale Klimasystem ist ein ausgesprochen komplexes und hochgradig nichtlineares System mit einer Vielzahl von Einflüssen und Interaktionen zwischen Variablen und Parametern. Komplementär zu der Beschreibung des Systems mit globalen Klimamodellen, kann Klima anhand der Interaktionsstruktur des Gesamtsystems durch Netzwerke beschrieben werden. Statt Details so genau wie möglich zu modellieren, werden hier Zeitreihendaten verwendet um zugrundeliegende Strukturen zu finden. Die Herausforderung liegt dann in der Interpretation dieser Strukturen. Um mich der Frage nach der Interpretation von Netzwerkmaßen zu nähern, suche ich nach einem allgemeinen Zusammenhang zwischen Eigenschaften der Netzwerktopologie und Eigenschaften des zugrundeliegenden physikalischen Systems. Dafür werden im Wesentlichen zwei Methoden entwickelt, die auf der Analyse von Temperaturentwicklungen gemäß der Advektions-Diffusions-Gleichung (ADE) basieren. Für die erste Methode wird die ADE mit offenen Randbedingungen und δ-peak Anfangsbedingungen gelöst. Die resultierenden lokalen Temperaturprofile werden verwendet um eine Korrelationsfunktion und damit ein Netzwerk zu definieren. Diese Netzwerke werden analysiert und mit Klimanetzen aus Daten verglichen. Die zweite Methode basiert auf der Diskretisierung der stochastischen ADE. Die resultierende lineare, stochastische Rekursionsgleichung wird verwendet um eine Korrelationsmatrix zu definieren, die nur von der Übergangsmatrix und der Varianz des stochastischen Störungsterms abhängt. Ich konstruiere gewichtete und ungewichtete Netzwerke für vier verschiedene Fälle und schlage Netzwerkmaße vor, die zwischen diesen Systemen zu unterscheiden helfen, wenn nur das Netzwerk und die Knotenpositionen gegeben sind. Die präsentierten Rekonstruktionsmethoden generieren Netzwerke, die konzeptionell und strukturell Klimanetzwerken ähneln und können somit als "proof of concept" der Methode der Klimanetzwerke, sowie als Interpretationshilfe betrachtet werden. / The earth’s climate is an extraordinarily complex, highly non-linear system with a multitude of influences and interactions between a very large number of variables and parameters. Complementary to the description of the system using global climate models, in recent years, a description based on the system’s interaction structure has been developed. Rather than modelling the system in as much detail as possible, here time series data is used to identify underlying large scale structures. The challenge then lies in the interpretation of these structures. In this thesis I approach the question of the interpretation of network measures from a general perspective, in order to derive a correspondence between properties of the network topology and properties of the underlying physical system. To this end I develop two methods of network construction from a velocity field, using the advection-diffusion-equation (ADE) for temperature-dissipation in the system. For the first method, the ADE is solved for δ-peak-shaped initial and open boundary conditions. The resulting local temperature profiles are used to define a correlation function and thereby a network. Those networks are analysed and compared to climate networks from data. Despite the simplicity of the model, it captures some of the most salient features of climate networks. The second network construction method relies on a discretisation of the ADE with a stochastic term. I construct weighted and unweighted networks for four different cases and suggest network measures, that can be used to distinguish between the different systems, based on the topology of the network and the node locations. The reconstruction methods presented in this thesis successfully model many features, found in climate networks from well-understood physical mechanisms. This can be regarded as a justification of the use of climate networks, as well as a tool for their interpretation.

Ce travail présente l’optimisation d’un système de spectroscopie Raman résolue en temps dédiée aux très hautes températures. Ce dispositif répond au besoin sans cesse croissant d’étudier en temps réels les transformations de phases et des cinétiques de réactions dans des environnements extrêmes. Ce dispositif a été validé dans des conditions d’usages sur des oxydes (Gd₂O3, Y₂O3, ZrO₂ , ZrSiO4 et CeO₂) et des nitrures (h-BN). Le potentiel du système a permis de lever les principaux verrous technologiques et instrumentaux qui limitent l’utilisation de la spectroscopie Raman à haute température. Les trois principaux faits marquants qui illustrent le caractère innovant de ce travail sont les suivants : Le premier correspond au nouveau record mondial d’analyse Raman à haute température à travers l’acquisition du mode E₂g de h-BN jusqu’à 2700°C. La comparaison des performances des deux voies Pockels et ICCD montrent que la voie Pockels est plus performante que l’ICCD, mais plus délicate de mise en oeuvre. Le second fait marquant concerne les autres applications du Raman résolu en temps, comme pour séparer la contribution de la diffusion Raman et de la luminescence. La dernière application quant à elle présente l’étude de la dépendance temporelle comparée des diffusions Raman résonnante et non résonante. Le Raman résonnant se déclenche systématiquement avant le non résonnant. Plus généralement, l’intérêt des méthodes Raman résolues en temps ouvre de nouveaux champs d’application dans la caractérisation de matériaux en condition extrêmes, éventuellement in situ : aéronautique, réfractaire ; sidérurgie, nucléaire, etc… / This work presents the optimization of a time-resolved Raman spectroscopy device dedicated to very high temperatures. This device meets the ever-increasing need to study in real time phase transformations and reaction kinetics in extreme environments. This device has been validated under working conditions on oxides (Gd₂O3, Y₂O3, ZrO₂ , ZrSiO4 et CeO₂) and nitrides (h-BN). The potentialities of the device have enabled the main technological and instrumental locks that limit the use of high temperature Raman spectroscopy to be removed. The three main highlights illustrating the innovative nature of this work are as follows. The first corresponds to the new world record for high temperature Raman analysis through the acquisition of the E₂g mode of h-BN up to 2700°C.A comparison of the performance of the two Pockels and ICCD channels shows that the Pockels channel is more efficient than the ICCD, but more difficult to implement. The second important fact concerns the other applications of time-resolved Raman, as to separate the contribution of Raman scattering and luminescence. The last application presents the study of the comparative time dependence of resonant and non-resonant Raman scattering. The resonant Raman is triggered systematically before the non-resonant. More generally, the interest of time-resolved Raman methods opens new fields of application in the characterization of materials in extreme conditions, possibly in situ: aeronautics, refractories, steel industry, nuclear, etc....

This thesis consists of a summary and four scientific articles. All four articles consider various aspects of stochastic differential equations and the purpose of the summary is to provide an introduction to this subject and to supply the notions required in order to fully understand the articles. In the first article we conduct a thorough study of the multi-dimensional Skorohod problem in time-dependent domains. In particular we prove the existence of cádlág solutions to the Skorohod problem with oblique reflection in time-independent domains with corners. We use this existence result to construct weak solutions to stochastic differential equations with oblique reflection in time-dependent domains. In the process of obtaining these results we also establish convergence results for sequences of solutions to the Skorohod problem and a number of estimates for solutions, with bounded jumps, to the Skorohod problem. The second article considers the problem of determining the sensitivities of a solution to a second order parabolic partial differential equation with respect to perturbations in the parameters of the equation. We derive an approximate representation of the sensitivities and an estimate of the discretization error arising in the sensitivity approximation. We apply these theoretical results to the problem of determining the sensitivities of the price of European swaptions in a LIBOR market model with respect to perturbations in the volatility structure (the so-called ‘Greeks’). The third article treats stopped diffusions in time-dependent graph domains with low regularity. We compare, numerically, the performance of one adaptive and three non-adaptive numerical methods with respect to order of convergence, efficiency and stability. In particular we investigate if the performance of the algorithms can be improved by a transformation which increases the regularity of the domain but, at the same time, reduces the regularity of the parameters of the diffusion. In the fourth article we use the existence results obtained in Article I to construct a projected Euler scheme for weak approximation of stochastic differential equations with oblique reflection in time-dependent domains. We prove theoretically that the order of convergence of the proposed algorithm is 1/2 and conduct numerical simulations which support this claim.

The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.

Wir untersuchen ein Defektkorrekturverfahren, das ein einfaches Upwind-Differenzenverfahren erster Ordnung mit einem zentralen Differenzenverfahren kombiniert, für ein- und zweidimensionale singulär gestörte Konvektions-Diffusions-Probleme auf einer Klasse von Shishkin-Typ-Gittern. Im eindimensionalen Fall wird nachgewiesen, dass das Verfahren von (fast) zweiter Ordnung, gleichmäßig bezüglich des Diffusionsparameters $\epsilon$ konvergiert. Zur Konvergenzanalyse für das zweidimensionale Modellproblem werden verschiedene Techniken diskutiert. In einem Spezialfall kann auf einem stückweise uniformen Shishkin-Gitter die $\epsilon$-gleichmäßige Konvergenz des Verfahrens von fast zweiter Ordnung gezeigt werden. Ferner sind die bisher bekannten Stabilitätsaussagen und ihre Verwendung zur Konvergenzanalysis der betrachteten Differenzenverfahren sowie Methoden zur Analyse von Defektkorrekturverfahren zusammengestellt. Einige Bemerkungen zu Defektkorrekturverfahren und Finite-Elemente-Methoden schließen die Arbeit ab. Numerische Experimente untermauern die theoretischen Resultate. / We consider a defect correction method that combines a first-order upwinded difference scheme with a second-order central difference scheme for model singularly perturbed convection-diffusion problems in one and two dimensions on a class of Shishkin-Type meshes. In one dimension, the method is shown to be convergent uniformly in the diffusion parameter $\epsilon$ of second order in the discrete maximum norm. To analyze the two-dimensional case, we discuss several proof techniques for defect correction methods. For a special problem with constant coefficients on a piecewise uniform Shishkin-mesh we can show the second order convergence of the considered scheme, uniformly with respect to the diffusion parameter. Moreover the known stability properties and their impact on the convergence analysis of the considered differnce schemes are compiled. Some remarks on defect correction and finite elements conclude the theses. Numerical experiments support our theoretical results.

No description available.

This is a book on numerical methods for singular perturbation problems - in particular stationary convection-dominated convection-diffusion problems. More precisely it is devoted to the construction and analysis of layer-adapted meshes underlying these numerical methods. An early important contribution towards the optimization of numerical methods by means of special meshes was made by N.S. Bakhvalov in 1969. His paper spawned a lively discussion in the literature with a number of further meshes being proposed and applied to various singular perturbation problems. However, in the mid 1980s this development stalled, but was enlivend again by G.I. Shishkin's proposal of piecewise- equidistant meshes in the early 1990s. Because of their very simple structure they are often much easier to analyse than other meshes, although they give numerical approximations that are inferior to solutions on competing meshes. Shishkin meshes for numerous problems and numerical methods have been studied since and they are still very much in vogue. With this contribution we try to counter this development and lay the emphasis on more general meshes that - apart from performing better than piecewise-uniform meshes - provide a much deeper insight in the course of their analysis. In this monograph a classification and a survey are given of layer-adapted meshes for convection-diffusion problems. It tries to give a comprehensive review of state-of-the art techniques used in the convergence analysis for various numerical methods: finite differences, finite elements and finite volumes. While for finite difference schemes applied to one-dimensional problems a rather complete convergence theory for arbitrary meshes is developed, the theory is more fragmentary for other methods and problems and still requires the restriction to certain classes of meshes.