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.

A Microscopia de Força Atômica é uma técnica que permite a aquisição de imagens em escalas nanométricas da superfície de quase todo tipo de material. Nessa escala, porém, as imagens podem apresentar uma relação sinal/ruído pobre, causado por efeitos degenerativos em sua qualidade. Para recuperar essas imagens ou minimizar os efeitos da degradação, técnicas de restauração de imagens são empregadas. Nas últimas décadas, diversas técnicas têm sido desenvolvidas e aplicadas com essa finalidade. Dentre elas, uma técnica de restauração, descrita aqui nesta tese, baseada na minimização de um funcional de Tikhonov com termos de regularização a um parâmetro, tem sido usada há alguns anos com resultados bastante satisfatórios no tratamento de imagens obtidas com o Microscópio de Força Atômica. O uso dessa técnica, entretanto, exige um grande esforço computacional que resulta em um tempo de execução elevado quando o programa que implementa o algoritmo de restauração é processado serialmente. Além disso, à medida que os equipamentos eletrônicos aumentam as suas capacidades, as imagens obtidas por esses equipamentos aumentam de resolução, assim como o esforço computacional e o tempo gasto para analisá-las e restaurálas. Assim, com o passar do tempo, o aumento da velocidade de processamento e do desempenho do programa de restauração tem-se tornado um problema cada vez mais crítico. Com o intuito de obter uma velocidade maior de processamento, nesta tese é descrita uma estratégia de implementação do algoritmo de restauração que faz uso de técnicas de computação paralela para se desenvolver uma nova versão paralela do programa de restauração. Os resultados obtidos com essa nova versão do programa mostram que a estratégia paralela adotada reduziu os tempos de execução e produziu bons desempenhos computacionais quando comparado com outras implementações feitas do mesmo algoritmo. Além disso, a nova estratégia apresenta níveis de desempenho maiores à medida que as resoluções das imagens restauradas aumentam, possibilitando a restauração de imagens maiores num tempo proporcionalmente mais curto.

A Microscopia de Força Atômica é uma técnica que permite a aquisição de imagens em escalas nanométricas da superfície de quase todo tipo de material. Nessa escala, porém, as imagens podem apresentar uma relação sinal/ruído pobre, causado por efeitos degenerativos em sua qualidade. Para recuperar essas imagens ou minimizar os efeitos da degradação, técnicas de restauração de imagens são empregadas. Nas últimas décadas, diversas técnicas têm sido desenvolvidas e aplicadas com essa finalidade. Dentre elas, uma técnica de restauração, descrita aqui nesta tese, baseada na minimização de um funcional de Tikhonov com termos de regularização a um parâmetro, tem sido usada há alguns anos com resultados bastante satisfatórios no tratamento de imagens obtidas com o Microscópio de Força Atômica. O uso dessa técnica, entretanto, exige um grande esforço computacional que resulta em um tempo de execução elevado quando o programa que implementa o algoritmo de restauração é processado serialmente. Além disso, à medida que os equipamentos eletrônicos aumentam as suas capacidades, as imagens obtidas por esses equipamentos aumentam de resolução, assim como o esforço computacional e o tempo gasto para analisá-las e restaurálas. Assim, com o passar do tempo, o aumento da velocidade de processamento e do desempenho do programa de restauração tem-se tornado um problema cada vez mais crítico. Com o intuito de obter uma velocidade maior de processamento, nesta tese é descrita uma estratégia de implementação do algoritmo de restauração que faz uso de técnicas de computação paralela para se desenvolver uma nova versão paralela do programa de restauração. Os resultados obtidos com essa nova versão do programa mostram que a estratégia paralela adotada reduziu os tempos de execução e produziu bons desempenhos computacionais quando comparado com outras implementações feitas do mesmo algoritmo. Além disso, a nova estratégia apresenta níveis de desempenho maiores à medida que as resoluções das imagens restauradas aumentam, possibilitando a restauração de imagens maiores num tempo proporcionalmente mais curto.

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.

Pass-by noise testing is an important step in vehicle design and regulation compliance. Finite element analysis simulations have been used to cut costs on prototyping and testing, but the high computational cost of simulating surface vibrations from complex geometries and the resulting airborne noise propagation is making the switch to digital twin methods not viable. This paper aims at investigating the use of equivalent source methods as an alternative to the before mentioned simulations. Through the use of a simple 2D model, the difficulties such as ill-conditioning of the transfer matrix and the required regularisation techniques such as TSVD and the Tikhonov L-curve method are tested and then applied to a mesh of a 3D engine model. Source and pressure field errors are measured and their origins are explained. A heavy emphasis is put on the model geometry as a source of error. Finally, rules of thumb based on the regularisation balance and the wavelength dependent pressure sampling positions are formulated in order to achieve usable results. / Bullerprovning vid passage är ett viktigt steg i fordonsdesign och regelefterlevnad. Simuleringar med finita elementanalyser har använts för att minska kostnaderna för prototypframtagning och provning, men de höga beräkningskostnaderna för att simulera ytvibrationer från komplexa geometrier och den resulterande luftburna bullerspridningen gör att övergången till digitala tvillingmetoder inte är genomförbar. Denna uppsats syftar till att undersöka användningen av ekvivalenta källmetoder som ett alternativ till de tidigare nämnda simuleringarna. Genom att använda en enkel 2D-modell testas svårigheterna som dålig konditionering av överföringsmatrisen och de nödvändiga regulariseringsteknikerna som TSVD och Tikhonov L-kurvmetoden och tillämpas sedan på ett nät av en 3D-motormodell. Käll- och tryckfältsfel mäts och deras ursprung förklaras. Stor vikt läggs vid modellgeometrin som en felkälla. Slutligen formuleras tumregler baserade på regulariseringsbalansen och de våglängdsberoende tryckprovtagningspositionerna för att uppnå användbara resultat.

The Gravity Recovery and Climate Experiment (GRACE) is a joint National Aeronautics and Space Administration / Deutsches Zentrum für Luftund Raumfahrt (NASA/DLR) mission to map the time-variable and mean gravity field of the Earth, and was launched on March 17, 2002. The nature of the gravity field inverse problem amplifies the noise in the data that creeps into the mid and high degree and order harmonic coefficients of the earth's gravity fields for monthly variability, making the GRACE estimation problem ill-posed. These errors, due to the use of imperfect models and data noise, are manifested as peculiar errors in the gravity estimates as north-south striping in the monthly global maps of equivalent water heights. In order to reduce these errors, this study develops a methodology based on Tikhonov regularization technique using the L-curve method in combination with orthogonal transformation method. L-curve is a popular aid for determining a suitable value of the regularization parameter when solving linear discrete ill-posed problems using Tikhonov regularization. However, the computational effort required to determine the L-curve can be prohibitive for a large scale problem like GRACE. This study implements a parameter-choice method, using Lanczos bidiagonalization that is a computationally inexpensive approximation to L-curve called L-ribbon. This method projects a large estimation problem on a problem of size of about two orders of magnitude smaller. Using the knowledge of the characteristics of the systematic errors in the GRACE solutions, this study designs a new regularization matrix that reduces the systematic errors without attenuating the signal. The regularization matrix provides a constraint on the geopotential coefficients as a function of its degree and order. The regularization algorithms are implemented in a parallel computing environment for this study. A five year time-series of the candidate regularized solutions show markedly reduced systematic errors without any reduction in the variability signal compared to the unconstrained solutions. The variability signals in the regularized series show good agreement with the hydrological models in the small and medium sized river basins and also show non-seasonal signals in the oceans without the need for post-processing. / text

Regression analysis refers to techniques for modeling and analyzing several variables in statistical learning. There are various types of regression models. In our study, we analyzed Generalized Partial Linear Models (GPLMs), which decomposes input variables into two sets, and additively combines classical linear models with nonlinear model part. By separating linear models from nonlinear ones, an inverse problem method Tikhonov regularization was applied for the nonlinear submodels separately, within the entire GPLM. Such a particular representation of submodels provides both a better accuracy and a better stability (regularity) under noise in the data. We aim to smooth the nonparametric part of GPLM by using a modified form of Multiple Adaptive Regression Spline (MARS) which is very useful for high-dimensional problems and does not impose any specific relationship between the predictor and dependent variables. Instead, it can estimate the contribution of the basis functions so that both the additive and interaction effects of the predictors are allowed to determine the dependent variable. The MARS algorithm has two steps: the forward and backward stepwise algorithms. In the rst one, the model is built by adding basis functions until a maximum level of complexity is reached. On the other hand, the backward stepwise algorithm starts with removing the least significant basis functions from the model. In this study, we propose to use a penalized residual sum of squares (PRSS) instead of the backward stepwise algorithm and construct PRSS for MARS as a Tikhonov regularization problem. Besides, we provide numeric example with two data sets / one has interaction and the other one does not have. As well as studying the regularization of the nonparametric part, we also mention theoretically the regularization of the parametric part. Furthermore, we make a comparison between Infinite Kernel Learning (IKL) and Tikhonov regularization by using two data sets, with the difference consisting in the (non-)homogeneity of the data set. The thesis concludes with an outlook on future research.

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.

The present thesis addresses both theoretical as well as numerical aspects of the treatment of nonlinear inverse problems. The first part considers Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. Sufficient as well as necessary conditions to establish convergence are introduced and convergence rate results are given for various parameter choice rules under a two sided nonlinearity constraint. Ultimately, both a posteriori as well as certain a priori parameter choice rules lead to identical converce rates. The theoretical results are supported and augmented by extensive numerical case studies. In particular it is shown, that the localization of the above mentioned nonlinearity constraint is not trivial. Incorrect localization will prevent convergence of the regularized to the exact solution. The second part of the thesis considers two open problems in inverse option pricing and electrical impedance tomography. While regularization through discretization is sufficient to overcome ill-posedness of the latter, the first requires a more sophisticated approach. It is shown, that the recovery of time dependent volatility and interest rate functions from observed option prices is everywhere locally ill-posed. This motivates Tikhonov-type or variational regularization with two parameters and penalty terms to simultaneously recover these functions. Two parameter choice rules using the L-hypersurface as well as a combination of L-curve and quasi-optimality are introduced. The results are again supported by extensive numerical case studies.