In the last decade l1-regularization became a powerful and popular tool for the regularization of Inverse Problems. While in the early years sparse solution were in the focus of research, recently also the case that the coefficients of the exact solution decay sufficiently fast was under consideration. In this paper we seek to show that l1-regularization is applicable and leads to optimal convergence rates even when the exact solution does not belong to l1 but only to l2. This is a particular example of over-smoothing regularization, i.e., the penalty implies smoothness properties the exact solution does not fulfill. We will make some statements on convergence also in this general context.

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For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems and medical imaging problems such as greenhouse gas tracking and computational tomography reconstruction. This dissertation describes advancements in computational tools for solving large-scale inverse problems and for uncertainty quantification. Oftentimes, inverse problems are ill-posed and large-scale. Iterative projection methods have dramatically reduced the computational costs of solving large-scale inverse problems, and regularization methods have been critical in obtaining stable estimations by applying prior information of unknowns via Bayesian inference. However, by combining iterative projection methods and variational regularization methods, hybrid projection approaches, in particular generalized hybrid methods, create a powerful framework that can maximize the benefits of each method. In this dissertation, we describe various advancements and extensions of hybrid projection methods that we developed to address three recent open problems. First, we develop hybrid projection methods that incorporate mixed Gaussian priors, where we seek more sophisticated estimations where the unknowns can be treated as random variables from a mixture of distributions. Second, we describe hybrid projection methods for mean estimation in a hierarchical Bayesian approach. By including more than one prior covariance matrix (e.g., mixed Gaussian priors) or estimating unknowns and hyper-parameters simultaneously (e.g., hierarchical Gaussian priors), we show that better estimations can be obtained. Third, we develop computational tools for a respirometry system that incorporate various regularization methods for both linear and nonlinear respirometry inversions. For the nonlinear systems, blind deconvolution methods are developed and prior knowledge of nonlinear parameters are used to reduce the dimension of the nonlinear systems. Simulated and real-data experiments of the respirometry problems are provided. This dissertation provides advanced tools for computational inversion and uncertainty quantification. / Doctor of Philosophy / For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems such as greenhouse gas tracking where the problem of estimating the amount of added or removed greenhouse gas at the atmosphere gets more difficult. The number of observations has been increased with improvements in measurement technologies (e.g., satellite). Therefore, the inverse problems become large-scale and they are computationally hard to solve. Another example of an inverse problem arises in tomography, where the goal is to examine materials deep underground (e.g., to look for gas or oil) or reconstruct an image of the interior of the human body from exterior measurements (e.g., to look for tumors). For tomography applications, there are typically fewer measurements than unknowns, which results in non-unique solutions. In this dissertation, we treat unknowns as random variables with prior probability distributions in order to compensate for a deficiency in measurements. We consider various additional assumptions on the prior distribution and develop efficient and robust numerical methods for solving inverse problems and for performing uncertainty quantification. We apply our developed methods to many numerical applications such as greenhouse gas tracking, seismic tomography, spherical tomography problems, and the estimation of CO2 of living organisms.

Cette thèse est consacrée à la recherche des zéros d'un opérateur maximal monotone structuré, à l'aide de systèmes dynamiques dissipatifs continus et discrets. Les solutions sont obtenues comme limites des trajectoires lorsque le temps t tend vers l'infini. On s'intéressera principalement aux dynamiques obtenues par régularisation de type Levenberg-Marquardt de la méthode de Newton. On décrira aussi les approches basées sur des dynamiques voisines.Dans un cadre Hilbertien, on s'intéresse à la recherche des zéros de l'opérateur maximal monotone structuré M = A + B, où A est un opérateur maximal monotone général et B est un opérateur monotone Lipschitzien. Nous introduisons des dynamiques continues et discrètes de type Newton régularisé faisant intervenir d'une façon séparée les résolvantes de l'opérateur A (implicites), et des évaluations de B (explicites). A l'aide de la représentation de Minty de l'opérateur A comme une variété Lipschitzienne, nous reformulons ces dynamiques sous une forme relevant du théorème de Cauchy-Lipschitz. Nous nous intéressons au cas particulier où A est le sous différentiel d'une fonction convexe, semi-continue inférieurement, et propre, et B est le gradient d'une fonction convexe, différentiable. Nous étudions le comportement asymptotique des trajectoires. Lorsque le terme de régularisation ne tend pas trop vite vers zéro, et en s'appuyant sur une analyse asymptotique de type Lyapunov, nous montrons la convergence des trajectoires. Par ailleurs, nous montrons la dépendance Lipschitzienne des trajectoires par rapport au terme de régularisation.Puis nous élargissons notre étude en considérant différentes classes de systèmes dynamiques visant à résoudre les inclusions monotones gouvernées par un opérateur maximal monotone structuré M = $partialPhi$+ B, où $partialPhi$ désigne le sous différentiel d'une fonction convexe, semicontinue inférieurement, et propre, et B est un opérateur monotone cocoercif. En s'appuyant sur une analyse asymptotique de type Lyapunov, nous étudions le comportement asymptotique des trajectoires de ces systèmes. La discrétisation temporelle de ces dynamiques fournit desalgorithmes forward-backward (certains nouveaux ).Finalement, nous nous intéressons à l'étude du comportement asymptotique des trajectoires de systèmes dynamiques de type Newton régularisé, dans lesquels on introduit un terme supplémentaire de viscosité évanescente de type Tikhonov. On obtient ainsi la sélection asymptotique d'une solution de norme minimale. / This thesis is devoted to finding zeroes of structured maximal monotone operators, by using discrete and continuous dissipative dynamical systems. The solutions are obtained as the limits of trajectories when the time t tends towards infinity.We pay special attention to the dynamics that are obtained by Levenberg-Marquardt regularization of Newton's method. We also revisit the approaches based on some related dynamical systems.In a Hilbert framework, we are interested in finding zeroes of a structured maximal monotone operator M = A + B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. We introduce discrete and continuous dynamical systems which are linked to Newton's method. They involve separately B and the resolvents of A, and are designed to splitting methods. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy-Lipschitz theorem. We focus on the particular case where A is the subdifferential of a convex lower semicontinuous proper function, and B is the gradient of a convex, continuously differentiable function. We study the asymptotic behavior of trajectories. When the regularization parameter does not tend to zero too rapidly, and by using Lyapunov asymptotic analysis, we show the convergence of trajectories. Besides, we show the Lipschitz continuous dependence of the solution with respect to the regularization term.Then we extend our study by considering various classes of dynamical systems which aim at solving inclusions governed by structured monotone operators M = $partialPhi$+ B, where $partialPhi$ is the subdifferential of a convex lower semicontinuous function, and B is a monotone cocoercive operator. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems. The time discretization of these dynamics gives various forward-backward splittingmethods (some new).Finally, we focus on the study of the asymptotic behavior of trajectories of the regularized Newton dynamics, in which we introduce an additional vanishing Tikhonov-like viscosity term.We thus obtain the asymptotic selection of the solution of minimal norm.

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.

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.

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.

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.