L'objectif de ce travail est de proposer une nouvelle démarche pour améliorer et accélérer les études de dimensionnement des pièces de turboréacteurs en avant-projets. Il s'agit de fournir une méthodologie complète pour la conception robuste sous contraintes. Cette méthodologie consiste en trois étapes : la réduction de la dimension et la méta-modélisation, la conception robuste sous contraintes puis la résolution de problèmes inverses sous contraintes. Ce sont les trois principaux sujets abordés dans cette thèse. La réduction de la dimension est un pré-traitement indispensable à toute étude. Son but est de ne conserver, pour une sortie choisie du système, que les entrées influentes. Ceci permet de réduire la taille du domaine d'étude afin de faciliter la compréhension du système et diminuer les temps de calculs des études. Les méthodes de méta-modélisations contribuent également à ces deux objectifs. L'idée est de remplacer le code de calculs coûteux par un modèle rapide à évaluer et qui représente bien la relation entre la sortie étudiée et les entrées du système. La conception robuste sous contraintes est une optimisation bi-objectifs où les différentes sources d'incertitudes du système sont prises en compte. Il s'agit, dans un premier temps, de recenser et modéliser les incertitudes puis de choisir une méthode de propagation de ces incertitudes dans le code de calculs. Ceci permet d'estimer les moments (moyenne et écart-type) de la loi de la sortie d'intérêt. L'optimisation de ces moments constitue les deux objectifs de la conception robuste. En dernier lieu, il s'agit de choisir la méthode d'optimisation multi-objectifs qui sera utilisée pour obtenir l'optimum robuste sous contraintes. La partie innovante de cette thèse porte sur le développement de méthodes pour la résolution de problèmes inverses mal posés. Ce sont des problèmes pour lesquels il peut y avoir une infinité de solutions constituant des ensembles non convexes et même disjoints. L'inversion a été considérée ici comme un complément à l'optimisation robuste dans laquelle l'optimum obtenu ne satisfaisait pas une des contraintes. Les méthodes d'inversion permettent alors de résoudre ce problème en trouvant plusieurs combinaisons des entrées qui satisfont la contrainte sous la condition de rester proche de l'optimum robuste. Le but est d'atteindre une valeur cible de la contrainte non satisfaite tout en respectant les autres contraintes du système auxquelles on ajoute la condition de proximité à l'optimum. Appliquée au dimensionnement d'un compresseur HP en avants-projets, cette méthodologie s'inscrit dans l'amélioration et l'accélération des études marquées par de nombreux rebouclages chronophages en termes de ressources informatiques et humaines. / The aim of this PhD dissertation is to propose a new approach to improve and accelerate preliminary design studies for turbofan engine components. This approach consists in a comprehensive methodology for robust design under constraints, following three stages : dimension reduction and metamodeling, robust design under constraints and finally inverse problem solving under constraints. These are the three main subjects of this PhD dissertation. Dimension reduction is an essential pre-processing for any study. Its aim is to keep only inputs with large effects on a selected output. This selection reduces the size of the domain on which is performed the study which reduces its computational cost and eases the (qualitative) understanding of the system of interest. Metamodeling also contributes to these two objectives by replacing the time-consuming computer code by a faster metamodel which approximates adequately the relationship between system inputs and the studied output. Robust design under constraints is a bi-objectives optimization where different uncertainty sources are included. First, uncertainties must be collected and modeled. Then a propagation method of uncertainties in the computation code must be chosen in order to estimate moments (mean and standard deviation) of output distribution. Optimization of these moments are the two robust design objectives. Finally, a multi-objectives optimization method has to be chosen to find a robust optimum under constraints. The development of methods to solve ill-posed inverse problems is the innovative part of this PhD dissertation. These problems can have infinitely many solutions constituting non convex or even disjoint sets. Inversion is considered here as a complement to robust design in the case where the obtained optimum doesn't satisfy one of the constraints. Inverse methods then enable to solve this problem by finding several input datasets which satisfy all the constraints and a condition of proximity to the optimum. The aim is to reach a target value of the unsatisfied constraint while respecting other system constraints and the optimum proximity condition. Applied to preliminary design of high pressure compressor, this methodology contributes to the improvement and acceleration of studies currently characterized by a numerous of loopbacks which are expensive in terms of cpu-time and human resources.

This paper is concerned with the numerical analysis of the autoconvolution equation $x*x=y$ restricted to the interval [0,1]. We present a discrete constrained least squares approach and prove its convergence in $L^p(0,1),1<p<\infinite$ , where the regularization is based on a prescribed bound for the total variation of admissible solutions. This approach includes the case of non-smooth solutions possessing jumps. Moreover, an adaption to the Sobolev space $H^1(0,1)$ and some remarks on monotone functions are added. The paper is completed by a numerical case study concerning the determination of non-monotone smooth and non-smooth functions x from the autoconvolution equation with noisy data y.

We prove some sufficient conditions for obtaining convergence rates in regularization of linear ill-posed problems in a Hilbert space setting and show that these conditions are directly related with the conditional stability in several concrete inverse problems for partial differential equations.

There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247--274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

Introducing a new method for measureing ultra-short laser pulses, the research group "Solid State Light Sources" of the Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Berlin, encountered a new type of autoconvolution problem. The so called SD-SPIDER method aims for the reconstruction of the real valued phase of a complex valued laser pulse from noisy measurements. The measurements are also complex valued and additionally influenced by a device-related kernel function. Although the autoconvolution equation has been examined intensively in the context of inverse problems, results for complex valued functions occurring as solutions and right-hand sides of the autoconvolution equation and for nontrivial kernels were missing. The thesis is a first step to bridge this gap. In the first chapter, the physical background is explained and especially the autoconvolution effect is pointed out. From this, the mathematical model is derived, leading to the final autoconvolution equation. Analytical results are given in the second chapter. It follows the numerical treatment of the problem in chapter three. A regularization approach is presented and tested with artificial data. In particular, a new parameter choice rule making use of a specific property of the SD-SPIDER method is proposed and numerically verified. / Bei der Entwicklung einer neuen Methode zur Messung ultra-kurzer Laserpulse stieß die Forschungsgruppe "Festkörper-Lichtquellen" des Max-Born-Institutes für Nichtlineare Optik und Kurzzeitspektroskopie, Berlin, auf ein neuartiges Selbstfaltungsproblem. Die so genannte SD-SPIDER-Methode dient der Rekonstruktion der reellen Phase eines komplexwertigen Laserpulses mit Hilfe fehlerbehafteter Messungen. Die Messwerte sind ebenfalls komplexwertig und zusätzlich beeinflusst von einer durch das Messprinzip erzeugten Kernfunktion. Obwohl Selbstfaltungsgleichungen intensiv im Kontext Inverser Probleme untersucht wurden, fehlen Resultate für komplexwertige Lösungen und rechte Seiten ebenso wie für nichttriviale Kernfunktionen. Die Diplomarbeit stellt einen ersten Schritt dar, diese Lücke zu schließen. Im ersten Kapitel wird der physikalische Hintergrund erläutert und insbesondere der Selbstfaltungseffekt erklärt. Davon ausgehend wird das mathematische Modell aufgestellt. Kapitel zwei befasst sich mit der Analysis der Gleichung. Es folgt die numerische Behandlung des Problems in Kapitel drei. Eine Regularisierungsmethode wird vorgestellt und an künstlichen Daten getestet. Insbesondere wird eine neue Regel zur Wahl des Regularisierungsparameters vorgeschlagen und numerisch bestätigt, welche auf einer speziellen Eigenschaft des SD-SPIDER Verfahrens beruht.

The authors study parameter choice strategies for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depend on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskii principle. For the convenience of the reader the authors review in an appendix a few instances where the validity of a variational inequality can be established.

Ill-posed inverse problems with quadratic structure are introduced, studied and solved. As an example an inverse problem appearing in laser optics is solved numerically based on a new regularized inversion algorithm. In addition, the theory of sparsity promoting regularization is extended to situations in which sparsity cannot be expected and also to equations with non-injective operators.

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