Background: This work presents a generalized technique to estimate pulmonary ventilation-to-volume (v/V) distributions using the multiple-breath nitrogen washout, in which both tidal volume (VT) and the end-expiratory lung volume (EELV) are allowed to vary during the maneuver. In addition, the volume of the series dead space (vd), unlike the classical model, is considered a common series unit connected to a set of parallel alveolar units. Methods: The numerical solution for simulated data, either error-free or with the N2 measurement contaminated with the addition of Gaussian random noise of 3 or 5 % standard deviation was tested under several conditions in a computational model constituted by 50 alveolar units with unimodal and bimodal distributions of v/V. Non-negative least squares regression with Tikhonov regularization was employed for parameter retrieval. The solution was obtained with either unconstrained or constrained (VT, EELV and vd) conditions. The Tikhonov gain was fixed or estimated and a weighting matrix (WM) was considered. The quality of estimation was evaluated by the sum of the squared errors (SSE) (between reference and recovered distributions) and by the deviations of the first three moments calculated for both distributions. Additionally, a shape classification method was tested to identify the solution as unimodal or bimodal, by counting the number of shape agreements after 1000 repetitions. Results: The accuracy of the results showed a high dependence on the noise amplitude. The best algorithm for SSE and moments included the constrained and the WM solvers, whereas shape agreement improved without WM, resulting in 97.2 % for unimodal and 90.0 % for bimodal distributions in the highest noise condition. Conclusions: In conclusion this generalized method was able to identify v/V distributions from a lung model with a common series dead space even with variable VT. Although limitations remain in presence of experimental noise, appropriate combination of processing steps were also found to reduce estimation errors.

La caractérisation d’objets enfouis à partir de mesures d’ondes diffractées est un problème présent dans de nombreuses applications comme l’exploration géophysique, le contrôle non-destructif, l’imagerie médicale, etc. Elle peut être obtenue numériquement par la résolution d’un problème inverse. Néanmoins, c’est un problème non linéaire et mal posé, ce qui rend la tâche difficile. Une reconstruction précise nécessite un choix judicieux de plusieurs paramètres très différents, dépendant des données de la méthode numérique d’optimisation choisie.La contribution principale de cette thèse est une étude de la reconstruction complète d’obstacles élastiques immergés à partir de mesures du champ lointain diffracté. Les paramètres à reconstruire sont la frontière, les coefficients de Lamé, la densité et la position de l’obstacle. On établit tout d’abord des résultats d’existence et d’unicité pour un problème aux limites généralisé englobant le problème direct d’élasto-acoustique. On analyse la sensibilité du champ diffracté par rapport aux différents paramètres du solide, ce qui nous conduit à caractériser les dérivées partielles de Fréchet comme des solutions du problème direct avec des seconds membres modifiés. Les dérivées sont calculées numériquement grâce à la méthode de Galerkine discontinue avec pénalité intérieure et le code est validé par des comparaisons avec des solutions analytiques. Ensuite, deux méthodologies sont introduites pour résoudre le problème inverse. Toutes deux reposent sur une méthode itérative de type Newton généralisée et la première consiste à retrouver les paramètres de nature différente indépendamment, alors que la seconde reconstruit tous les paramètre en même temps. À cause du comportement différent des paramètres, on réalise des tests de sensibilité pour évaluer l’influence de ces paramètres sur les mesures. On conclut que les paramètres matériels ont une influence plus faible sur les mesures que les paramètres de forme et, ainsi, qu’une stratégie efficace pour retrouver des paramètres de nature distincte doit prendre en compte ces différents niveaux de sensibilité. On a effectué de nombreuses expériences à différents niveaux de bruit, avec des données partielles ou complètes pour retrouver certains paramètres, par exemple les coefficients de Lamé et les paramètres de forme, la densité, les paramètres de forme et la localisation. Cet ensemble de tests contribue à la mise en place d’une stratégie pour la reconstruction complète des conditions plus proches de la réalité. Dans la dernière partie de la thèse, on étend ces résultats à des matériaux plus complexes, en particulier élastiques anisotropes. / The characterization of hidden objects from scattered wave measurements arises in many applications such as geophysical exploration, non destructive testing, medical imaging, etc. It can be achieved numerically by solving an Inverse Problem. However, this is a nonlinear and ill-posed problem, thus a difficult task. A successful reconstruction requires careful selection of very different parameters depending on the data and the chosen optimization numerical method.The main contribution of this thesis is an investigation of the full reconstruction of immersed elastic scatterers from far-field pattern measurements. The sought-after parameters are the boundary, the Lamé coefficients, the density and the location of the obstacle. First, existence and uniqueness results of a generalized Boundary Value Problem including the direct elasto-acoustic problem are established. The sensitivity of the scattered field with respect to the different parametersdescribing the solid is analyzed and we end up with the characterization of the corresponding partial Fréchet derivatives as solutions to the direct problem with modified right-hand sides. These Fréchet derivatives are computed numerically thanks to the Interior Penalty Discontinuous Galerkin method and the code is validated thanks to comparison with analytical solutions. Then, two solution methodologies are introduced for solving the inverse problem. Both are based on an iterative regularized Newton-type methodology and the first one consists in retrieving the parameters of different nature independently, while the second one reconstructs all parameters together. Due to the different behavior of the parameters, sensitivity tests are performed to assess the impact of the parameters on the measurements. We conclude that material parameters have a weaker influence on the measurements than shape parameters, and therefore, a successful strategy to retrieve parameters of distinct nature should take into account these different levels of sensitivity. Various experiments at different noise levels and with full or limited aperture data are carried out to retrieve some of the physical properties, e.g. Lamé coefficients with shape parameters, density with shape parameters a, density, shape and location. This set of tests contributes to a final strategy for the full reconstruction and in more realistic conditions. In the final part of the thesis, we extend the results to more complex material parameters, in particular anisotropic elastic.

No description available.

Le progrès technologique nécessite des techniques de plus en plus sophistiquées et précises de traitement de matériaux. Nous étudions le traitement de matériaux par faisceaux de haute énergie : un jet d’eau abrasif, une sonde ionique focalisée, un laser. L’évolution de la surface du matériau sous l’action du faisceau de haute énergie est modélisée par une EDP. Cette équation contient l’ensemble des coefficients inconnus - les paramètres de calibration de mo- dèle. Les paramètres inconnus peuvent être calibrés par minimisation de la fonction coût, c’est-à-dire, la fonction qui décrit la différence entre le résultat de la modélisation et les données expérimentales. Comme la surface modélisée est une solution du problème d’EDP, cela rentre dans le cadre de l’optimisation sous contrainte d’EDP. L’identification a été rendue bien posée par la régularisation du type Tikhonov. Le gradient de la fonction coût a été obtenu en utilisant les deux méthodes : l’approche adjointe et la différen- ciation automatique. Une fois la fonction coût et son gradient obtenus, nous avons utilisé un minimiseur L-BFGS pour réaliser la minimisation.Le problème de la non-unicité de la solution a été résolu pour le problème de traitement par le jet d’eau abrasif. Des effets secondaires ne sont pas inclus dans le modèle. Leur impact sur le procédé de calibration a été évité. Ensuite, le procédé de calibration a été validé pour les données synthétiques et expérimentales. Enfin, nous avons proposé un critère pour distinguer facilement entre le régime thermique et non- thermique d’ablation par laser. / The technological progress demands more and more sophisticated and precise techniques of the treatment of materials. We study the machining of the material with the high energy beams: the abrasive waterjet, the focused ion beam and the laser. Although the physics governing the energy beam interaction with material is very different for different application, we can use the same approach to the mathematical modeling of these processes.The evolution of the material surface under the energy beam impact is modeled by PDE equation. This equation contains a set of unknown parameters - the calibration parameters of the model. The unknown parameters can be identified by minimization of the cost function, i.e., function that describes the differ- ence between the result of modeling and the corresponding experimental data. As the modeled surface is a solution of the PDE problem, this minimization is an example of PDE-constrained optimization problem. The identification problem was regularized using Tikhonov regularization. The gradient of the cost function was obtained both by using the variational approach and by means of the automatic differentiation. Once the cost function and its gradient calculated, the minimization was performed using L-BFGS minimizer.For the abrasive waterjet application the problem of non-uniqueness of numerical solution is solved. The impact of the secondary effects non included into the model is avoided as well. The calibration procedure is validated on both synthetic and experimental data.For the laser application, we presented a simple criterion that allows to distinguish between the thermal and non-thermal laser ablation regimes.

Background: This work presents a generalized technique to estimate pulmonary ventilation-to-volume (v/V) distributions using the multiple-breath nitrogen washout, in which both tidal volume (VT) and the end-expiratory lung volume (EELV) are allowed to vary during the maneuver. In addition, the volume of the series dead space (vd), unlike the classical model, is considered a common series unit connected to a set of parallel alveolar units. Methods: The numerical solution for simulated data, either error-free or with the N2 measurement contaminated with the addition of Gaussian random noise of 3 or 5 % standard deviation was tested under several conditions in a computational model constituted by 50 alveolar units with unimodal and bimodal distributions of v/V. Non-negative least squares regression with Tikhonov regularization was employed for parameter retrieval. The solution was obtained with either unconstrained or constrained (VT, EELV and vd) conditions. The Tikhonov gain was fixed or estimated and a weighting matrix (WM) was considered. The quality of estimation was evaluated by the sum of the squared errors (SSE) (between reference and recovered distributions) and by the deviations of the first three moments calculated for both distributions. Additionally, a shape classification method was tested to identify the solution as unimodal or bimodal, by counting the number of shape agreements after 1000 repetitions. Results: The accuracy of the results showed a high dependence on the noise amplitude. The best algorithm for SSE and moments included the constrained and the WM solvers, whereas shape agreement improved without WM, resulting in 97.2 % for unimodal and 90.0 % for bimodal distributions in the highest noise condition. Conclusions: In conclusion this generalized method was able to identify v/V distributions from a lung model with a common series dead space even with variable VT. Although limitations remain in presence of experimental noise, appropriate combination of processing steps were also found to reduce estimation errors.

Malgré les progrès accomplis dans la réduction du bruit des ventilateurs axiaux subsoniques, le bruit de raie basse fréquence, composante gênante du bruit, demeure difficile à contrôler. Aux basses fréquences, l'ajout de matériaux absorbants est inefficace et les silencieux encombrants et coûteux. Il faut alors faire appel à des "mesures curatives", comme le contrôle actif acoustique ou le contrôle actif à la source. Les contributions de ces travaux de doctorat sont : 1) l'estimation des sources aéroacoustiques du bruit de raie par modèle inverse, 2) une méthode de contrôle actif nécessitant un seul haut-parleur et 3) une méthode de contrôle passif adaptatif basée sur l'ajout d'obstructions dans l'écoulement. Les développements théoriques menés dans cette thèse sont valides pour les ventilateurs axiaux subsoniques et les expériences ont été réalisées sur un ventilateur de radiateur d'automobile.

Tikhonov-type regularization of linear and nonlinear ill-posed problems in abstract spaces under sparsity constraints gained relevant attention in the past years. Since under some weak assumptions all regularized solutions are sparse if the l1-norm is used as penalty term, the l1-regularization was studied by numerous authors although the non-reflexivity of the Banach space l1 and the fact that such penalty functional is not strictly convex lead to serious difficulties. We consider the case that the sparsity assumption is narrowly missed. This means that the solutions may have an infinite number of nonzero but fast decaying components. For that case we formulate and prove convergence rates results for the l1-regularization of nonlinear operator equations. In this context, we outline the situations of Hölder rates and of an exponential decay of the solution components.

Uma área de intensa atividade de pesquisa em nanotecnologia está relacionada à restauração de imagens obtidas através dos microscópios de força atômica (AFM). Durante o processo de aquisição as imagens de AFM estão sujeitas às limitações impostas pela instrumentação utilizada e costumam apresentar pobres relações sinal/ruído, bem como a manifestação dos efeitos degenerativos relacionados à interação entre o tip e a amostra. Buscando tratar imagens obtidas de estruturas biológicas, tem sido propostos e testados diversos algoritmos de restauração, e nesta dissertação é utilizado um método baseado na Regularização de Tikhonov. Ao longo do desenvolvimento de diversos algoritmos de restauração foi observado que os indicadores matemáticos usuais para avaliação de desempenho não correspondiam à percepção visual relativa às imagens restauradas. Este trabalho propõe e avalia diferentes indicadores de desempenho para algoritmos de restauração utilizados no pósprocessamento de imagens obtidas em escala nanométrica por microscopia de força atômica e imagens artificiais criadas especialmente para testar o indicador de desempenho utilizado. São investigados indicadores usuais empregados em restauração de imagens, sendo também proposta a implementação de indicadores baseado nas distâncias de Bregman. O trabalho foi dividido em duas partes. Primeiramente foi gerado e restaurado um conjunto de imagens utilizando o funcional de regularização de Tikhonov, sendo distribuídas entre 50 pessoas que foram chamadas de avaliadores humanos. A tarefa consistia em avaliar as imagens, procurando identificar o quanto as imagens restauradas se aproximavam das imagens originais, no que se relaciona à visão humana. A segunda parte consistiu na implementação de uma rotina computacional para o cálculo de vários indicadores matemáticos. Posteriormente foi realizada a comparação dos resultados obtidos através dos indicadores matemáticos com a percepção visual dos avaliadores humanos, sendo esta última essencialmente qualitativa, sendo então transformada em uma escala numérica para permitir então a comparação com os indicadores matemáticos. Observou-se que dependendo dos parâmetros utilizados no algoritmo de restauração baseado no funcional de regularização de Tikhonov e dos tipos de imagens restauradas, os indicadores baseados nas Distâncias de Bregman apresentam melhores resultados do que aqueles obtidos com indicadores tradicionais, como por exemplo o erro médio quadrático (MSE Mean Square Error), levando a uma concordância maior destes resultados quando comparados com a avaliação feita pelos avaliadores humanos. / The restoration of images acquired with Atomic Force Microscopes (AFM) consists on a very intense research area in nanotechnology. During the acquisition process the AFM images are subjected to the constraints imposed by the experimental apparatus usually presenting poor signal to noise ratios as well as the effects of tip-sample interaction. In order to process images related to biological structures, several restoration algorithms have been proposed and tested, and in the present dissertation is used a method based in Tikhonovs regularization. Throughout the development of several restoration algorithms it has been observed that the usual performance measures did not correspond to the human visual perception of the restored images. In the present dissertation are proposed and evaluated several performance measures for restoration algorithms used in the post-processing of images acquired in nanoscale with AFM as well as artificial images created in order to test the performance measure used. Usually used performance measures are investigated as well as measures based on Bregman distances. The research was conducted in two parts. First a set of images was generated and restored using Tikhonovs regularization functional, being then distributed to 50 human referees. Their task consisted in evaluating the images in order to grade how close the restored images were to the original images, in respect to human visual perception. The second part of the research consisted in the implementation of a computational program to perform the computation of the mathematical performance measures. Afterwards a comparison was performed of the mathematical performance measures with respect to the visual perception of the human referees, being the latter essentially qualitative which was then transformed in a numerical scale in order to allow the comparison with the mathematical measures. It was observed that depending on the parameters used in the restoration algorithm based in Tikhonovs regularization functional and the type of the restored image, the mathematical performance measures based on Bregman distances present better results than those obtained with other usual performance measures such as the Mean Square Error (MSE) yielding to a better agreement with the evaluation performed by the human referees.

Uma área de intensa atividade de pesquisa em nanotecnologia está relacionada à restauração de imagens obtidas através dos microscópios de força atômica (AFM). Durante o processo de aquisição as imagens de AFM estão sujeitas às limitações impostas pela instrumentação utilizada e costumam apresentar pobres relações sinal/ruído, bem como a manifestação dos efeitos degenerativos relacionados à interação entre o tip e a amostra. Buscando tratar imagens obtidas de estruturas biológicas, tem sido propostos e testados diversos algoritmos de restauração, e nesta dissertação é utilizado um método baseado na Regularização de Tikhonov. Ao longo do desenvolvimento de diversos algoritmos de restauração foi observado que os indicadores matemáticos usuais para avaliação de desempenho não correspondiam à percepção visual relativa às imagens restauradas. Este trabalho propõe e avalia diferentes indicadores de desempenho para algoritmos de restauração utilizados no pósprocessamento de imagens obtidas em escala nanométrica por microscopia de força atômica e imagens artificiais criadas especialmente para testar o indicador de desempenho utilizado. São investigados indicadores usuais empregados em restauração de imagens, sendo também proposta a implementação de indicadores baseado nas distâncias de Bregman. O trabalho foi dividido em duas partes. Primeiramente foi gerado e restaurado um conjunto de imagens utilizando o funcional de regularização de Tikhonov, sendo distribuídas entre 50 pessoas que foram chamadas de avaliadores humanos. A tarefa consistia em avaliar as imagens, procurando identificar o quanto as imagens restauradas se aproximavam das imagens originais, no que se relaciona à visão humana. A segunda parte consistiu na implementação de uma rotina computacional para o cálculo de vários indicadores matemáticos. Posteriormente foi realizada a comparação dos resultados obtidos através dos indicadores matemáticos com a percepção visual dos avaliadores humanos, sendo esta última essencialmente qualitativa, sendo então transformada em uma escala numérica para permitir então a comparação com os indicadores matemáticos. Observou-se que dependendo dos parâmetros utilizados no algoritmo de restauração baseado no funcional de regularização de Tikhonov e dos tipos de imagens restauradas, os indicadores baseados nas Distâncias de Bregman apresentam melhores resultados do que aqueles obtidos com indicadores tradicionais, como por exemplo o erro médio quadrático (MSE Mean Square Error), levando a uma concordância maior destes resultados quando comparados com a avaliação feita pelos avaliadores humanos. / The restoration of images acquired with Atomic Force Microscopes (AFM) consists on a very intense research area in nanotechnology. During the acquisition process the AFM images are subjected to the constraints imposed by the experimental apparatus usually presenting poor signal to noise ratios as well as the effects of tip-sample interaction. In order to process images related to biological structures, several restoration algorithms have been proposed and tested, and in the present dissertation is used a method based in Tikhonovs regularization. Throughout the development of several restoration algorithms it has been observed that the usual performance measures did not correspond to the human visual perception of the restored images. In the present dissertation are proposed and evaluated several performance measures for restoration algorithms used in the post-processing of images acquired in nanoscale with AFM as well as artificial images created in order to test the performance measure used. Usually used performance measures are investigated as well as measures based on Bregman distances. The research was conducted in two parts. First a set of images was generated and restored using Tikhonovs regularization functional, being then distributed to 50 human referees. Their task consisted in evaluating the images in order to grade how close the restored images were to the original images, in respect to human visual perception. The second part of the research consisted in the implementation of a computational program to perform the computation of the mathematical performance measures. Afterwards a comparison was performed of the mathematical performance measures with respect to the visual perception of the human referees, being the latter essentially qualitative which was then transformed in a numerical scale in order to allow the comparison with the mathematical measures. It was observed that depending on the parameters used in the restoration algorithm based in Tikhonovs regularization functional and the type of the restored image, the mathematical performance measures based on Bregman distances present better results than those obtained with other usual performance measures such as the Mean Square Error (MSE) yielding to a better agreement with the evaluation performed by the human referees.

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.