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We revisit in L2-spaces the autoconvolution equation x ∗ x = y with solutions which are real-valued or complex-valued functions x(t) defined on a finite real interval, say t ∈ [0,1]. Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low order convergence rates for regularized solutions of nonlinear ill-posed operator equations F(x)=y with solutions x† in a Hilbert space setting. So for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of F (x† )∗ fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on [0,2] and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex L2-functions with a fixed and uniformly bounded modulus function.

The focus of this paper is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in the literature by a couple different methods. In this paper we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We are going to work out the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capability of our method is illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.

The development of highly coherent X-ray sources offers new possibilities to image biological structures at different scales exploiting the refraction of X-rays. The coherence properties of the third-generation synchrotron radiation sources enables efficient implementations of phase contrast techniques. One of the first measurements of the intensity variations due to phase contrast has been reported in 1995 at the European Synchrotron Radiation Facility (ESRF). Phase imaging coupled to tomography acquisition allows threedimensional imaging with an increased sensitivity compared to absorption CT. This technique is particularly attractive to image samples with low absorption constituents. Phase contrast has many applications, ranging from material science, paleontology, bone research to medicine and biology. Several methods to achieve X-ray phase contrast have been proposed during the last years. In propagation based phase contrast, the measurements are made at different sample-to-detector distances. While the intensity data can be acquired and recorded, the phase information of the signal has to be "retrieved" from the modulus data only. Phase retrieval is thus an illposed nonlinear problem and regularization techniques including a priori knowledge are necessary to obtain stable solutions. Several phase recovery methods have been developed in recent years. These approaches generally formulate the phase retrieval problem as a linear one. Nonlinear treatments have not been much investigated. The main purpose of this work was to propose and evaluate new algorithms, in particularly taking into account the nonlinearity of the direct problem. In the first part of this work, we present a Landweber type nonlinear iterative scheme to solve the propagation based phase retrieval problem. This approach uses the analytic expression of the Fréchet derivative of the phase-intensity relationship and of its adjoint, which are presented in detail. We also study the effect of projection operators on the convergence properties of the method. In the second part of this thesis, we investigate the resolution of the linear inverse problem with an iterative thresholding algorithm in wavelet coordinates. In the following, the two former algorithms are combined and compared with another nonlinear approach based on sparsity regularization and a fixed point algorithm. The performance of theses algorithms are evaluated on simulated data for different noise levels. Finally the algorithms were adapted to process real data sets obtained in phase CT at the ESRF at Grenoble.

CNPQ; LASCA / Este trabalho propõe um algoritmo para identificar descontinuidades internas em uma peça metálica a partir de sinais A-scan provenientes de um ensaio não-destrutivo por ultrassom. As descontinuidades são identificadas por seu formato, dimensões, orientação e localização. Esse algoritmo é baseado na resolução de problemas inversos. O problema da identificação de descontinuidades é modelado utilizando uma abordagem inédita, em que a descontinuidade é caracterizada por dois elementos distintos: a sua amplitude de espalhamento e a localização de seu centro. Com essa nova forma de representação, o problema de identificação da descontinuidade passa a ser dividido em dois subproblemas. O primeiro é o problema de reconstrução para encontrar a localização do centro da descontinuidade e o segundo é o problema de caracterização para estimar os parâmetros geométricos da descontinuidade. Os ensaios realizados comprovam que o problema de localização é resolvido de forma satisfatória utilizando o algoritmo de reconstrução de imagens esparsas UTSR (ultrasonic sparse reconstruction). A caracterização da descontinuidade é realizada por um algoritmo baseado no método dos mínimos quadrados não-lineares, com a inclusão de um termo de regularização não-quadrático utilizando norma l1. Os resultados obtidos tanto com dados simulados como com dados experimentais mostram que esse algoritmo estima as descontinuidades de forma satisfatória. / The present work proposes an algorithm to identify internal discontinuities in metallic specimen from A-scan ultrasonic signals. Discontinuities are identified by their shape, size, orientation and location. This algorithm is based on inverse problems. The identification of discontinuities is modeled using a novel approach. The discontinuity is modeled by two distinct elements: its scattering amplitude and its center location. Through this approach, the identification problem is split into two sub problems. The first is a reconstruction problem to find the location of the discontinuity. The second is a model identification problem to estimate geometric parameters of the discontinuity. Simulations and experimental validation show that the localization problem is satisfactorily solved using the UTSR (ultrasonic sparse reconstruction) algorithm. The characterization of discontinuities is carried out by an algorithm based on non-linear least squares with l1 norm regularization. The results obtained with simulated data as well with experimental data show a fine estimation of discontinuities.

Au cours des dernières années, s’est développé un intérêt tout particulier pour l’optimisation sans dérivée. Ce domaine de recherche se divise en deux catégories: une déterministe et l’autre stochastique. Bien qu’il s’agisse du même domaine, peu de liens ont déjà été établis entre ces deux branches. Cette thèse a pour objectif de combler cette lacune, en montrant comment les techniques issues de l’optimisation déterministe peuvent améliorer la performance des stratégies évolutionnaires, qui font partie des meilleures méthodes en optimisation stochastique. Sous certaines hypothèses, les modifications réalisées assurent une forme de convergence globale, c’est-à-dire une convergence vers un point stationnaire de premier ordre indépendamment du point de départ choisi. On propose ensuite d’adapter notre algorithme afin qu’il puisse traiter des problèmes avec des contraintes générales. On montrera également comment améliorer les performances numériques des stratégies évolutionnaires en incorporant un pas de recherche au début de chaque itération, dans laquelle on construira alors un modèle quadratique utilisant les points où la fonction coût a déjà été évaluée. Grâce aux récents progrès techniques dans le domaine du calcul parallèle, et à la nature parallélisable des stratégies évolutionnaires, on propose d’appliquer notre algorithme pour résoudre un problème inverse d’imagerie sismique. Les résultats obtenus ont permis d’améliorer la résolution de ce problème. / In recent years, there has been significant and growing interest in Derivative-Free Optimization (DFO). This field can be divided into two categories: deterministic and stochastic. Despite addressing the same problem domain, only few interactions between the two DFO categories were established in the existing literature. In this thesis, we attempt to bridge this gap by showing how ideas from deterministic DFO can improve the efficiency and the rigorousness of one of the most successful class of stochastic algorithms, known as Evolution Strategies (ES’s). We propose to equip a class of ES’s with known techniques from deterministic DFO. The modified ES’s achieve rigorously a form of global convergence under reasonable assumptions. By global convergence, we mean convergence to first-order stationary points independently of the starting point. The modified ES’s are extended to handle general constrained optimization problems. Furthermore, we show how to significantly improve the numerical performance of ES’s by incorporating a search step at the beginning of each iteration. In this step, we build a quadratic model using the points where the objective function has been previously evaluated. Motivated by the recent growth of high performance computing resources and the parallel nature of ES’s, an application of our modified ES’s to Earth imaging Geophysics problem is proposed. The obtained results provide a great improvement for the problem resolution.

L'intérêt de l'échantillonnage compressé dans l'imagerie ultrasonore a été récemment évalué largement par plusieurs équipes de recherche. Suite aux différentes configurations d'application, il a été démontré que les données RF peuvent être reconstituées à partir d'un faible nombre de mesures et / ou en utilisant un nombre réduit d'émission d'impulsions ultrasonores. Selon le modèle de l'échantillonnage compressé, la résolution des images ultrasonores reconstruites à partir des mesures compressées dépend principalement de trois aspects: la configuration d'acquisition, c.à.d. l'incohérence de la matrice d'échantillonnage, la régularisation de l'image, c.à.d. l'a priori de parcimonie et la technique d'optimisation. Nous nous sommes concentrés principalement sur les deux derniers aspects dans cette thèse. Néanmoins, la résolution spatiale d'image RF, le contraste et le rapport signal sur bruit dépendent de la bande passante limitée du transducteur d'imagerie et du phénomène physique lié à la propagation des ondes ultrasonores. Pour surmonter ces limitations, plusieurs techniques de traitement d'image en fonction de déconvolution ont été proposées pour améliorer les images ultrasonores. Dans cette thèse, nous proposons d'abord un nouveau cadre de travail pour l'imagerie ultrasonore, nommé déconvolution compressée, pour combiner l'échantillonnage compressé et la déconvolution. Exploitant une formulation unifiée du modèle d'acquisition directe, combinant des projections aléatoires et une convolution 2D avec une réponse impulsionnelle spatialement invariante, l'avantage de ce cadre de travail est la réduction du volume de données et l'amélioration de la qualité de l'image. Une méthode d'optimisation basée sur l'algorithme des directions alternées est ensuite proposée pour inverser le modèle linéaire, en incluant deux termes de régularisation exprimant la parcimonie des images RF dans une base donnée et l'hypothèse statistique gaussienne généralisée sur les fonctions de réflectivité des tissus. Nous améliorons les résultats ensuite par la méthode basée sur l'algorithme des directions simultanées. Les deux algorithmes sont évalués sur des données simulées et des données in vivo. Avec les techniques de régularisation, une nouvelle approche basée sur la minimisation alternée est finalement développée pour estimer conjointement les fonctions de réflectivité des tissus et la réponse impulsionnelle. Une investigation préliminaire est effectuée sur des données simulées. / The interest of compressive sampling in ultrasound imaging has been recently extensively evaluated by several research teams. Following the different application setups, it has been shown that the RF data may be reconstructed from a small number of measurements and/or using a reduced number of ultrasound pulse emissions. According to the model of compressive sampling, the resolution of reconstructed ultrasound images from compressed measurements mainly depends on three aspects: the acquisition setup, i.e. the incoherence of the sampling matrix, the image regularization, i.e. the sparsity prior, and the optimization technique. We mainly focused on the last two aspects in this thesis. Nevertheless, RF image spatial resolution, contrast and signal to noise ratio are affected by the limited bandwidth of the imaging transducer and the physical phenomenon related to Ultrasound wave propagation. To overcome these limitations, several deconvolution-based image processing techniques have been proposed to enhance the ultrasound images. In this thesis, we first propose a novel framework for Ultrasound imaging, named compressive deconvolution, to combine the compressive sampling and deconvolution. Exploiting an unified formulation of the direct acquisition model, combining random projections and 2D convolution with a spatially invariant point spread function, the benefit of this framework is the joint data volume reduction and image quality improvement. An optimization method based on the Alternating Direction Method of Multipliers is then proposed to invert the linear model, including two regularization terms expressing the sparsity of the RF images in a given basis and the generalized Gaussian statistical assumption on tissue reflectivity functions. It is improved afterwards by the method based on the Simultaneous Direction Method of Multipliers. Both algorithms are evaluated on simulated and in vivo data. With regularization techniques, a novel approach based on Alternating Minimization is finally developed to jointly estimate the tissue reflectivity function and the point spread function. A preliminary investigation is made on simulated data.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / A formulação explícita matricial desenvolvida nesta tese de doutorado foi proposta visando ser uma alternativa na solução de Problemas Inversos de estimativa de propriedades radiativas em meios participantes homogêneos unidimensionais usando a Equação de Transferência Radiativa para modelar a interação da radiação com o meio participante. A equação de transporte é formulada em forma matricial e o domínio angular é discretizado usando conceitos do método de ordenadas discretas e a expansão da função de fase do espalhamento anisotrópico em uma série de polinômios de Legendre. A formulação proposta consiste em uma formulação explícita para o problema inverso. Um arranjo apropriado das condições de contorno prescritas (fluxos incidentes) e dos fluxos emergentes nos contornos de uma placa permitem o cálculo direto do operador de transmissão, do operador albedo e do operador de colisão. A partir do operador de colisão calculado são obtidos os valores estimados dos coeficientes de extinção total e de espalhamento. São apresentadas as formulações para problemas em regime estacionário e em regime transiente, bem como os resultados para alguns casos-teste. / The explicit matrix formulation developed in the present thesis has been proposed as an alternative for the solution of Inverse Problems for radiative properties estimation in one-dimensional homogeneous participating media using Radiative transfer equation for the modeling of the radiation interaction with the participating medium. This transport equation is formulated in a matrix form and the angular domain is discretized using concepts of the discrete ordinates methods and the expansion of the function of phase function of anisotropic scattering in a series of Legendre polynomial. The formulation proposed consists on an explicit formulation for the inverse problem. An adequate assembly of the prescribed boundary conditions (incidents flux) and of the emerging flux at the boundaries of the slab allows the direct computation of the transmission, albedo and collision operators. From the computed collision operator estimated values for total extinction and scattering coefficients are obtained. The formulations for steady state and transient situations are presented, as well as test case results.

A espectroscopia fotoacústica é uma técnica muito utilizada na caracterização de materiais. Ela é baseada no efeito fotoacústico, que consiste na geração de som a partir da incidência de radiação eletromagnética intermitente numa amostra confinada numa pequena câmara fechada e preenchida com algum gás. O efeito fotoacústico depende de muitos parâmetros como a freqüência de modulação da radiação eletromagnética incidente e de parâmetros ópticos e térmicos do material em estudo. O modelo de Rosencwaig-Gersho (RG) tem sido empregado na estimativa de parâmetros ópticos e térmicos da amostra utilizando os casos especiais de opacidade e de comportamento térmico do material. Há o interesse em utilizar o modelo RG para a estimativa de dois ou mais parâmetros, simultaneamente. Houve algum sucesso com a aplicação do método inverso utilizando dados experimentais sintéticos, ou seja, utilizando dados gerados pelo modelo RG (método direto), aos quais adiciona-se ruído com o intuito de simular dados experimentais. O objetivo do presente trabalho é a utilização de dados experimentais reais nesta abordagem do método inverso. Para isso foi realizado um projeto de experimento onde se estudou a possibilidade de estimação simultânea de parâmetros a partir das condições gerais do experimento real. Neste projeto de experimento, o estudo dos coeficientes de sensibilidade modificados desempenha um papel central. Desta forma, foi elaborado um algoritmo para o projeto de experimento, onde é estudada a relação entre tais coeficientes e a estimação dos parâmetros feita pelo método inverso. Como exemplo, é apresentada a execução de um projeto de experimento simples, realizado qualitativamente, porém capaz de apontar para um experimento fotoacústico que fornecerá dados experimentais reais para estimativa de um determinado parâmetro físico da amostra. Com esses dados experimentais o problema inverso foi aplicado para a estimação da difusividade térmica da amostra. Por fim, uma discussão é feita com relação ao valor da difusividade térmica obtida pelo método inverso e com relação à abordagem do método inverso na espectroscopia fotoacústica. / Photoacoustic spectroscopy is a very effective technique for optical and thermal characterization of materials. This technique is based on photoacoustic effect. It occurs when a material sample placed inside a closed cell filled with any gas is illuminated with periodically interrupted light. This effect depends on many parameters as modulation frequency of the incident electromagnetic radiation and on optical and thermal properties of the material studied. The Rosencwaig-Gersho (RG) model has been used for parameter estimation where special cases of opaqueness and thermal behavior of the material are considered. One has interest in the use of the general RG model to estimate two or more parameters, simultaneously. In this way, the inverse problem approach has been applied with the direct problem modeled with the general RG model. However, synthetic experimental data has been used in such approach, e.g., noise is added on the data, obtained from direct problem, in order to simulate real experimental data. The main goal of this work is to use the real experimental data in the inverse problem approach. It was needed to carry out the sensitivity study of the parameters with respect to the overall experimental conditions. The modified sensitivity coefficients play an important role in such study. Therefore, an algorithm was built up to study the relationship between modified sensitivity coefficients for the successful inverse approach of parameter estimation. As an example, a simple and qualitative study of sensitivity of the parameters was done, using such algorithm. It pointed out the conditions of the real experimental arrangement in which experimental data were obtained to estimate a unique parameter of the sample. With these experimental data the inverse problem was applied for determining the thermal diffusivity of the sample. Finally, a discussion is presented on the value obtained for the thermal diffusivity by this approach and on the application of the inverse method on the photoacoustic spectroscopy.