Cette thèse porte sur le développement de méthodes de Monte-Carlo pour calculer des représentations Feynman-Kac impliquant des opérateurs sous forme divergence avec un coefficient de diffusion constant par morceaux. Les méthodes proposées sont des variantes de la marche sur les sphères à l'intérieur des zones avec un coefficient de diffusion constant et des techniques de différences finies stochastiques pour traiter les conditions aux interfaces aussi bien que les conditions aux limites de différents types. En combinant ces deux techniques, on obtient des marches aléatoires dont le score calculé le long du chemin fourni un estimateur biaisé de la solution de l'équation aux dérivées partielles considérée. On montre que le biais global de notre algorithme est en général d'ordre deux par rapport au pas de différences finies. Ces méthodes sont ensuite appliquées au problème direct lié à la tomographie par impédance électrique pour la détection de tumeurs. Une technique de réduction de variance est également proposée dans ce cadre. On traite finalement du problème inverse de la détection de tumeurs à partir de mesures de surfaces à l'aide de deux algorithmes stochastiques basés sur une représentation paramétrique de la tumeur ou des tumeurs sous forme d'une ou plusieurs sphères. De nombreux essais numériques sont proposés et montrent des résultats probants dans la localisation des tumeurs. / This thesis deals with the development of Monte-Carlo methods to compute Feynman-Kac representations involving divergence form operators with a piecewise constant diffusion coefficient. The proposed methods are variations around the walk on spheres method inside the regions with a constant diffusion coefficient and stochastic finite differences techniques to treat the interface conditions as well as the different kinds of boundary conditions. By combining these two techniques, we build random walks which score computed along the walk gives us a biased estimator of the solution of the partial differential equation we consider. We prove that the global bias is in general of order two with respect to the finite difference step. These methods are then applied for tumour detection to the forward problem in electrical impedance tomography. A variance reduction technique is also proposed in this case. Finally, we treat the inverse problem of tumours detection from surface measurements using two stochastics algorithms based on a spherical parametric representation of the tumours. Many numerical tests are proposed and show convincing results in the localization of the tumours.

This thesis deals with a design of a power supply for a measuring equipment for a geophysical mapping intended to do a reconstruction of an electrical impedance of a ground environment of water reservoirs. The power source is going to be used for laboratory measurements and for outdoor experiments. The thesis describes the selection of suitable components, equipment design and testing. The theoretical part explains the functions of used components and the principle of an electrical impedance tomography. The conclusion summarizes the properties of the proposed solution and evaluates the final product.

Disertační práce, jež má teoretický charakter, je zaměřena na vylepšení a výzkum algoritmů pro zobrazování vnitřní struktury vodivých objektů, hlavně biologických tkání a orgánů pomocí elektrické impedanční tomografie (EIT). V práci je formulován teoretický rámec EIT. Dále jsou prezentovány a porovnány algoritmy pro řešení inverzní úlohy, které zajišťují efektivní rekonstrukci prostorového rozložení elektrických vlastností ve zkoumaném objektu a jejích zobrazení. Hlavní myšlenka vylepšeného algoritmu, který je založen na deterministickém přístupu, spočívá v zavedení dodatečných technik: level set a nebo fuzzy filtru. Kromě toho, je ukázána metoda 2-D rekonstrukce rozložení konduktivity z jediného komponentu magnetického pole a to konkrétně z-tové složky magnetického toku. Byly vytvořeny numerické modely biologické tkáně s určitým rozložení admitivity (nebo konduktivity) pro otestování těchto algoritmů. Výsledky získané z rekonstrukcí pomocí vylepšených algoritmů jsou ukázány a porovnány.

<div>Laser-to-composite interactions are becoming increasingly common in diverse applications such as diagnostics, fabrication and machining, and weapons systems. Lasers are capable of not only performing non-contact diagnostics, but also inducing seemingly imperceptible structural damage to materials. In safety-critical venues like aerospace, automotive, and civil infrastructure where composites are playing an increasingly prominent role, it is desirable to have means of sensing laser exposure on a composite material. Self-sensing materials may be a powerful method of addressing this need. Herein, we present an exploratory study on the potential of using changes in electrical measurements as a way of detecting laser exposure to a carbon nanofiber (CNF)-modified glass fiber/epoxy laminate. CNFs were dispersed in liquid epoxy resin prior to laminate fabrication via hand layup. The dispersed CNFs form a three-dimensional conductive network which allows for electrical measurements to be taken from the traditionally insulating glass fiber/epoxy material system. It is expected that damage to the network will disrupt the electrical pathways, thereby causing the material to exhibit slightly higher resistance. To test laser sensing capabilities, a resistance baseline of the CNF-modified glass fiber/epoxy specimens was first established before laser exposure. These specimens were then exposed to an infra-red laser operating at 1064 nm, 35 kHz, and pulse duration of 8 ns. The specimens were irradiated for a total of 20 seconds (4 exposures each at 5 seconds). The resistances of the specimens were then measured again post-ablation. In this study, it was found that for 1.0 wt.% CNF by weight the average resistance increased by about 18 percent. However, this values varied for specimens with different weight fractions. This established that the laser was indeed causing damage to the specimen sufficient to evoke a change in electrical properties. In order to expand on this result, electrical impedance tomography (EIT) was employed for localization of laser exposures of 1, 3, and 5 seconds on a larger specimen, a 3.25” square plate. EIT was used to measure the changes in conductivity after each exposure. EIT was not only successful in detecting damage that was virtually imperceptible to the human-eye, but it also accurately localized the exposure sites. The post-ablation conductivity of the exposure sites decreased in a manner that was comparable to the resistance increase obtained during prior testing. Based on this preliminary study, this research could lead to the development of a real-time exposure detection and tracking system for the measurement, fabrication, and defense industries.</div>

La capacité de la force de Lorentz à relier un déplacement mécanique à un courant électrique présente un intérêt certain pour l'acoustique médicale, et trois applications ont été étudiées dans cette thèse. Dans la première partie de ce travail, un hydrophone a été développé pour effectuer des champs de vitesse acoustique. Cet hydrophone était constitué d'un fil de cuivre vibrant dans un champ magnétique. Un modèle a été élaboré pour déterminer une relation entre la pression acoustique et le courant électrique mesure, qui est induit par force de Lorentz lorsque le fil vibre dans un champ acoustique. Un prototype a ensuite été conçu et sa résolution spatiale, sa réponse fréquentielle, sa sensibilité, sa résistance et sa réponse directionnelle ont été examinées. Une méthode d'imagerie appelée Tomographie d'Impedance Electrique par Force de Lorentz a aussi été étudiée. Dans cette méthode, un tissu biologique est déplacé par ultrasons dans un champ magnétique, ce qui induit un courant électrique par force de Lorentz. L'impédance électrique du tissu peut ensuite être déduite de la mesure du courant. Cette technique a été appliquée pour réaliser des images d'un fantôme de gélatine, d'un muscle de bœuf, et d'une lésion thermique dans un échantillon de poulet. Cela a montré que la méthode est potentiellement utile pour fournir un contraste supplémentaire à des images ultrasonores classiques. Enfin, cette thèse a démontré que des ondes de cisaillement peuvent être générées dans des tissus mous par force de Lorentz. Cela a été réalisé en appliquant un courant électrique par deux électrodes dans un solide mou place dans un champ magnétique. Des ondes de cisaillement ont été observées dans des échantillons de gélatine et de foie. La vitesse des ondes de cisaillement a été utilisée pour calculer l'élasticité et leur source pour cartographier la conductivité électrique des échantillons / The ability of the Lorentz force to link a mechanical displacement to an electrical current presents a strong interest for medical acoustics, and three applications were studied in this thesis. In the first part of this work, a hydrophone was developed for mapping the particle velocity of an acoustic field. This hydrophone was constructed using a thin copper wire and an external magnetic field. A model was elaborated to determine the relationship between the acoustic pressure and the measured electrical current, which is induced by Lorentz force when the wire vibrates in the acoustic field of an ultrasound transducer. The built prototype was characterized and its spatial resolution, frequency response, sensitivity, robustness and directivity response were investigated. An imaging method called Lorentz Force Electrical Impedance Tomography was also studied. In this method, a biological tissue is vibrated by ultrasound in a magnetic field, which induces an electrical current by Lorentz force. The electrical impedance of the tissue can be deduced from the measurement of the current. This technique was applied for imaging a gelatin phantom, a beef muscle sample, and a thermal lesion in a chicken breast sample. This showed the method may be useful for providing additional contrast to conventional ultrasound imaging. Finally, this thesis demonstrated that shear waves can be generated in soft tissues using Lorentz force. This work was performed by applying an electrical current with two electrodes in a soft solid placed in a magnetic field. Shear waves were observed in gelatin phantom and liver sample. The speed of the shear waves were used to compute elasticity and their source to map the electrical conductivity of the samples

This doctoral thesis proposes a new algorithm for the reconstruction of impedance images in monitored objects. The algorithm eliminates the spatial resolution problems present in existing reconstruction methods, and, with respect to the monitored objects, it exploits both the partial knowledge of configuration and the material composition. The discussed novel method is designed to recognize certain significant fields of interest, such as material defects or blood clots and tumors in biological images. The actual reconstruction process comprises two phases; while the former stage is focused on industry-related images, with the aim to detect defects in conductive materials, the latter one concentrates on biomedical applications. The thesis also presents a description of the numerical model used to test the algorithm. The testing procedure was centred on the resulting impedivity value, influence of the regularization parameter, initial value of the numerical model impedivity, and effect exerted by noise on the voltage electrodes upon the overall reconstruction results. Another issue analyzed herein is the possibility of reconstructing impedance images from components of the magnetic flux density measured outside the investigated object. The given magnetic field is generated by a current passing through the object. The created algorithm for the reconstruction of impedance images is modeled on the proposed algorithm for EIT-based reconstruction of impedance images from voltage. The algoritm was tested for stability, influence of the regularization parameter, and initial conductivity. From the general perspective, the thesis describes the methodology for both magnetic field measurement via NMR and processing of the obtained data.