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Numerous scientific applications have seen the rise of massive inverse problems, where there are too much data to implement an all-at-once strategy to compute a solution. Additionally, tools for regularizing ill-posed inverse problems are infeasible when the problem is too large. This thesis focuses on the development of row-action methods, which can be used to iteratively solve inverse problems when it is not possible to access the entire data-set or forward model simultaneously. We investigate these techniques for linear inverse problems and for separable, nonlinear inverse problems where the objective function is nonlinear in one set of parameters and linear in another set of parameters. For the linear problem, we perform a convergence analysis of these methods, which shows favorable asymptotic and initial convergence properties, as well as a trade-off between convergence rate and precision of iterates that is based on the step-size. These row-action methods can be interpreted as stochastic Newton and stochastic quasi-Newton approaches on a reformulation of the least squares problem, and they can be analyzed as limited memory variants of the recursive least squares algorithm. For ill-posed problems, we introduce sampled regularization parameter selection techniques, which include sampled variants of the discrepancy principle, the unbiased predictive risk estimator, and the generalized cross-validation. We demonstrate the effectiveness of these methods using examples from super-resolution imaging, tomography reconstruction, and image classification. / Doctor of Philosophy / Numerous scientific problems have seen the rise of massive data sets. An example of this is super-resolution, where many low-resolution images are used to construct a high-resolution image, or 3-D medical imaging where a 3-D image of an object of interest with hundreds of millions voxels is reconstructed from x-rays moving through that object. This work focuses on row-action methods that numerically solve these problems by repeatedly using smaller samples of the data to avoid the computational burden of using the entire data set at once. When data sets contain measurement errors, this can cause the solution to get contaminated with noise. While there are methods to handle this issue, when the data set becomes massive, these methods are no longer feasible. This dissertation develops techniques to avoid getting the solution contaminated with noise, even when the data set is immense. The methods developed in this work are applied to numerous scientific applications including super-resolution imaging, tomography, and image classification.

Ce travail traite de deux problèmes de grande importances en géodésie physique. Le premier porte sur la détermination du géoïde sur une zone terrestre donnée. Si la terre était une sphère homogène, la gravitation en un point, serait entièrement déterminée à partir de sa distance au centre de la terre, ou de manière équivalente, en fonction de son altitude. Comme la terre n'est ni sphérique ni homogène, il faut calculer en tout point la gravitation. A partir d'un ellipsoïde de référence, on cherche la correction à apporter à une première approximation du champ de gravitation afin d'obtenir un géoïde, c'est-à-dire une surface sur laquelle la gravitation est constante. En fait, la méthode utilisée est la méthode de collocation par moindres carrés qui sert à résoudre des grands problèmes aux moindres carrés généralisés. Le seconde partie de cette thèse concerne un problème inverse géodésique qui consiste à trouver une répartition de masses ponctuelles (caractérisées par leurs intensités et positions), de sorte que le potentiel généré par eux, se rapproche au maximum d'un potentiel donné. Sur la terre entière une fonction potentielle est généralement exprimée en termes d'harmoniques sphériques qui sont des fonctions de base à support global la sphère. L'identification du potentiel cherché se fait en résolvant un problème aux moindres carrés. Lorsque seulement une zone limitée de la Terre est étudiée, l'estimation des paramètres des points masses à l'aide des harmoniques sphériques est sujette à l'erreur, car ces fonctions de base ne sont plus orthogonales sur un domaine partiel de la sphère. Le problème de la détermination des points masses sur une zone limitée est traitée par la construction d'une base de Slepian qui est orthogonale sur le domaine limité spécifié de la sphère. Nous proposons un algorithme itératif pour la résolution numérique du problème local de détermination des masses ponctuelles et nous donnons quelques résultats sur la robustesse de ce processus de reconstruction. Nous étudions également la stabilité de ce problème relativement au bruit ajouté. Nous présentons quelques résultats numériques ainsi que leurs interprétations. / This work focuses on the study of two well-known problems in physical geodesy. The first problem concerns the determination of the geoid on a given area on the earth. If the Earth were a homogeneous sphere, the gravity at a point would be entirely determined from its distance to the center of the earth or in terms of its altitude. As the earth is neither spherical nor homogeneous, we must calculate gravity at any point. From a reference ellipsoid, we search to find the correction to a mathematical approximation of the gravitational field in order to obtain a geoid, i.e. A surface on which gravitational potential is constant. The method used is the method of least squares collocation which is the best for solving large generalized least squares problems. In the second problem, We are interested in a geodetic inverse problem that consists in finding a distribution of point masses (characterized by their intensities and positions), such that the potential generated by them best approximates a given potential field. On the whole Earth a potential function is usually expressed in terms of spherical harmonics which are basis functions with global support. The identification of the two potentials is done by solving a least-squares problem. When only a limited area of the Earth is studied, the estimation of the point-mass parameters by means of spherical harmonics is prone to error, since they are no longer orthogonal over a partial domain of the sphere. The point-mass determination problem on a limited region is treated by the construction of a Slepian basis that is orthogonal over the specified limited domain of the sphere. We propose an iterative algorithm for the numerical solution of the local point mass determination problem and give some results on the robustness of this reconstruction process. We also study the stability of this problem against added noise. Some numerical tests are presented and commented.

The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a variant of the discrepancy principle is analyzed. In many cases such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.

The dissertation suggests a generalized version of Tikhonov regularization and analyzes its properties. The focus is on convergence rates theory and an extensive example for regularization with Poisson distributed data is given.

Les dynamiques des systèmes modélisés par des équations aux dérivées partielles (EDPs) en dimension infinie sont largement liées aux réseaux physiques. La synthèse de la commande et l'analyse de la stabilité de ces systèmes sont étudiées dans cette thèse. Les systèmes singulièrement perturbés, contenant des échelles de temps multiples sont naturels dans les systèmes physiques avec des petits paramètres parasitaires, généralement de petites constantes de temps, les masses, les inductances, les moments d'inertie. La théorie des perturbations singulières a été introduite pour le contrôle à la fin des années $1960$, son assimilation dans la théorie du contrôle s'est rapidement développée et est devenue un outil majeur pour l'analyse et la synthèse de la commande des systèmes. Les perturbations singulières sont une façon de négliger la transition rapide, en la considérant dans une échelle de temps rapide séparée. Ce travail de thèse se concentre sur les systèmes hyperboliques linéaires avec des échelles de temps multiples modélisées par un petit paramètre de perturbation. Tout d'abord, nous étudions une classe de systèmes hyperboliques linéaires singulièrement perturbés. Comme le système contient deux échelles de temps, en mettant le paramètre de la perturbation à zéro, deux sous-systèmes, le système réduit et la couche limite, sont formellement calculés. La stabilité du système complet de lois de conservation implique la stabilité des deux sous-systèmes. En revanche un contre-exemple est utilisé pour illustrer que la stabilité des deux sous-systèmes ne suffit pas à garantir la stabilité du système complet. Cela montre une grande différence avec ce qui est bien connu pour les systèmes linéaires en dimension finie modélisés par des équations aux dérivées ordinaires (EDO). De plus, sous certaines conditions, l'approximation de Tikhonov est obtenue pour tels systèmes par la méthode de Lyapunov. Plus précisément, la solution de la dynamique lente du système complet est approchée par la solution du système réduit lorsque le paramètre de la perturbation est suffisamment petit. Deuxièmement, le théorème de Tikhonov est établi pour les systèmes hyperboliques linéaires singulièrement perturbés de lois d'équilibre où les vitesses de transport et les termes sources sont à la fois dépendant du paramètre de la perturbation ainsi que les conditions aux bords. Sous des hypothèses sur la continuité de ces termes et sous la condition de la stabilité, l'estimation de l'erreur entre la dynamique lente du système complet et le système réduit est obtenue en fonction de l'ordre du paramètre de la perturbation. Troisièmement, nous considérons des systèmes EDO-EDP couplés singulièrement perturbés. La stabilité des deux sous-systèmes implique la stabilité du système complet où le paramètre de la perturbation est introduit dans la dynamique de l'EDP. D'autre part, cela n'est pas valable pour le système où le paramètre de la perturbation est présent dans l'EDO. Le théorème Tikhonov pour ces systèmes EDO-EDP couplés est prouvé par la technique de Lyapunov. Enfin, la synthèse de la commande aux bords est abordée en exploitant la méthode des perturbations singulières. Le système réduit converge en temps fini. La synthèse du contrôle aux bords est mise en œuvre pour deux applications différentes afin d'illustrer les résultats principaux de ce travail. / Systems modeled by partial differential equations (PDEs) with infinite dimensional dynamics are relevant for a wide range of physical networks. The control and stability analysis of such systems become a challenge area. Singularly perturbed systems, containing multiple time scales, often occur naturally in physical systems due to the presence of small parasitic parameters, typically small time constants, masses, inductances, moments of inertia. Singular perturbation was introduced in control engineering in late $1960$s, its assimilation in control theory has rapidly developed and has become a tool for analysis and design of control systems. Singular perturbation is a way of neglecting the fast transition and considering them in a separate fast time scale. The present thesis is concerned with a class of linear hyperbolic systems with multiple time scales modeled by a small perturbation parameter. Firstly we study a class of singularly perturbed linear hyperbolic systems of conservation laws. Since the system contains two time scales, by setting the perturbation parameter to zero, the two subsystems, namely the reduced subsystem and the boundary-layer subsystem, are formally computed. The stability of the full system implies the stability of both subsystems. However a counterexample is used to illustrate that the stability of the two subsystems is not enough to guarantee the full system's stability. This shows a major difference with what is well known for linear finite dimensional systems. Moreover, under certain conditions, the Tikhonov approximation for such system is achieved by Lyapunov method. Precisely, the solution of the slow dynamics of the full system is approximated by the solution of the reduced subsystem for sufficiently small perturbation parameter. Secondly the Tikhonov theorem is established for singularly perturbed linear hyperbolic systems of balance laws where the transport velocities and source terms are both dependent on the perturbation parameter as well as the boundary conditions. Under the assumptions on the continuity for such terms and under the stability condition, the estimate of the error between the slow dynamics of the full system and the reduced subsystem is the order of the perturbation parameter. Thirdly, we consider singularly perturbed coupled ordinary differential equation ODE-PDE systems. The stability of both subsystems implies that of the full system where the perturbation parameter is introduced into the dynamics of the PDE system. On the other hand, this is not true for system where the perturbation parameter is presented to the ODE. The Tikhonov theorem for such coupled ODE-PDE systems is proved by Lyapunov technique. Finally, the boundary control synthesis is achieved based on singular perturbation method. The reduced subsystem is convergent in finite time. Boundary control design to different applications are used to illustrate the main results of this work.

Cauchy problems for elliptic partial differential equations arise in many important applications, such as, cardiography, nondestructive testing, heat transfer, sonic boom produced by a maneuvering aerofoil, etc. Elliptic Cauchy problems are typically ill-posed, i.e., there may not be a solution for some Cauchy data, and even if a solution exists uniquely, it may not depend continuously on the Cauchy data. The ill-posedness causes numerical instability and makes the classical numerical methods inappropriate to solve such problems. For Cauchy problems, the research on uniqueness, stability, and efficient numerical methods are of significant interest to mathematicians. The main focus of this thesis is to develop numerical techniques for elliptic Cauchy problems. Elliptic Cauchy problems can be approached as data completion problems, i.e., from over-specified Cauchy data on an accessible part of the boundary, one can try to recover missing data on the inaccessible part of the boundary. Then, the Cauchy problems can be solved by finding a so-lution to a well-posed boundary value problem for which the recovered data constitute a boundary condition on the inaccessible part of the boundary. In this thesis, we use natural linearization approach to transform the linear Cauchy problem into a problem of solving a linear operator equation. We consider this operator in a weaker image space H−1, which differs from the previous works where the image space of the operator is usually considered as L2 . The lower smoothness of the image space will make a problem a bit more ill-posed. But under such settings, we can prove the compactness of the considered operator. At the same time, it allows a relaxation of the assumption concerning noise. The numerical methods that can cope with these ill-posed operator equations are the so called regularization methods. One prominent example of such regularization methods is Tikhonov regularization which is frequently used in practice. Tikhonov regularization can be considered as a least-squares tracking of data with a regularization term. In this thesis we discuss a possibility to improve the reconstruction accuracy of the Tikhonov regularization method by using an iterative modification of Tikhonov regularization. With this iterated Tikhonov regularization the effect of the penalty term fades away as iterations go on. In the application of iterated Tikhonov regularization, we find that for severely ill-posed problems such as elliptic Cauchy problems, discretization has such a powerful influence on the accuracy of the regularized solution that only with some reasonable discretization level, desirable accuracy can be achieved. Thus, regularization by projection method which is commonly known as self-regularization is also considered in this thesis. With this method, the regularization is achieved only by discretization along with an appropriate choice of discretization level. For all regularization methods, the choice of an appropriate regularization parameter is a crucial issue. For this purpose, we propose the balancing principle which is a recently introduced powerful technique for the choice of the regularization parameter. While applying this principle, a balance between the components related to the convergence rate and stability in the accuracy estimates has to be made. The main advantage of the balancing principle is that it can work in an adaptive way to obtain an appropriate value of the regularization parameter, and it does not use any quantitative knowledge of convergence rate or stability. The accuracy provided by this adaptive strategy is worse only by a constant factor than one could achieve in the case of known stability and convergence rates. We apply the balancing principle in both iterated Tikhonov regularization and self-regularization methods to choose the proper regularization parameters. In the thesis, we also investigate numerical techniques based on iterative Tikhonov regular-ization for nonlinear elliptic Cauchy problems. We consider two types of problems. In the first kind, the nonlinear problem can be transformed to a linear problem while in the second kind, linearization of the nonlinear problem is not possible, and for this we propose a special iterative method which differs from methods such as Landweber iteration and Newton-type method which are usually based on the calculation of the Frech´et derivative or adjoint of the equation. Abundant examples are presented in the thesis, which illustrate the performance of the pro-posed regularization methods as well as the balancing principle. At the same time, these examples can be viewed as a support for the theoretical results achieved in this thesis. In the end of this thesis, we describe the sonic boom problem, where we first encountered the ill-posed nonlinear Cauchy problem. This is a very difficult problem and hence we took this problem to provide a motivation for the model problems. These model problems are discussed one by one in the thesis in the increasing order of difficulty, ending with the nonlinear problems in Chapter 5. The main results of the dissertation are communicated in the article [35].

The dissertation suggests a generalized version of Tikhonov regularization and analyzes its properties. The focus is on convergence rates theory and an extensive example for regularization with Poisson distributed data is given.

The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a variant of the discrepancy principle is analyzed. In many cases such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.

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.