We discuss a full-waveform based material profile reconstruction in two-dimensional heterogeneous semi-infinite domains. In particular, we try to image the spatial variation of shear moduli/wave velocities, directly in the time-domain, from scant surficial measurements of the domain's response to prescribed dynamic excitation. In addition, in one-dimensional media, we try to image the spatial variability of elastic and attenuation properties simultaneously. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries, and adopt perfectly-matched-layers (PMLs) as the boundary wave absorbers. Within this framework we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs for transient scalar wave simulations in heterogeneous semi-infinite domains. We use, as is typically done, complex-coordinate stretching transformations in the frequency-domain, and recover the governing PDEs in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error. To tackle the inversion, we adopt a PDE-constrained optimization approach, that formally leads to a classic KKT (Karush-Kuhn-Tucker) system comprising an initial-value state, a final-value adjoint, and a time-invariant control problem. We iteratively update the velocity profile by solving the KKT system via a reduced space approach. To narrow the feasibility space and alleviate the inherent solution multiplicity of the inverse problem, Tikhonov and Total Variation (TV) regularization schemes are used, endowed with a regularization factor continuation algorithm. We use a source frequency continuation scheme to make successive iterates remain within the basin of attraction of the global minimum. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on both one- and two-dimensional examples, including the Marmousi benchmark problem, that lead efficiently to the reconstruction of heterogeneous profiles involving both horizontal and inclined layers, as well as of inclusions within layered systems. / text

Le progrès technologique nécessite des techniques de plus en plus sophistiquées et précises de traitement de matériaux. Nous étudions le traitement de matériaux par faisceaux de haute énergie : un jet d’eau abrasif, une sonde ionique focalisée, un laser. L’évolution de la surface du matériau sous l’action du faisceau de haute énergie est modélisée par une EDP. Cette équation contient l’ensemble des coefficients inconnus - les paramètres de calibration de mo- dèle. Les paramètres inconnus peuvent être calibrés par minimisation de la fonction coût, c’est-à-dire, la fonction qui décrit la différence entre le résultat de la modélisation et les données expérimentales. Comme la surface modélisée est une solution du problème d’EDP, cela rentre dans le cadre de l’optimisation sous contrainte d’EDP. L’identification a été rendue bien posée par la régularisation du type Tikhonov. Le gradient de la fonction coût a été obtenu en utilisant les deux méthodes : l’approche adjointe et la différen- ciation automatique. Une fois la fonction coût et son gradient obtenus, nous avons utilisé un minimiseur L-BFGS pour réaliser la minimisation.Le problème de la non-unicité de la solution a été résolu pour le problème de traitement par le jet d’eau abrasif. Des effets secondaires ne sont pas inclus dans le modèle. Leur impact sur le procédé de calibration a été évité. Ensuite, le procédé de calibration a été validé pour les données synthétiques et expérimentales. Enfin, nous avons proposé un critère pour distinguer facilement entre le régime thermique et non- thermique d’ablation par laser. / The technological progress demands more and more sophisticated and precise techniques of the treatment of materials. We study the machining of the material with the high energy beams: the abrasive waterjet, the focused ion beam and the laser. Although the physics governing the energy beam interaction with material is very different for different application, we can use the same approach to the mathematical modeling of these processes.The evolution of the material surface under the energy beam impact is modeled by PDE equation. This equation contains a set of unknown parameters - the calibration parameters of the model. The unknown parameters can be identified by minimization of the cost function, i.e., function that describes the differ- ence between the result of modeling and the corresponding experimental data. As the modeled surface is a solution of the PDE problem, this minimization is an example of PDE-constrained optimization problem. The identification problem was regularized using Tikhonov regularization. The gradient of the cost function was obtained both by using the variational approach and by means of the automatic differentiation. Once the cost function and its gradient calculated, the minimization was performed using L-BFGS minimizer.For the abrasive waterjet application the problem of non-uniqueness of numerical solution is solved. The impact of the secondary effects non included into the model is avoided as well. The calibration procedure is validated on both synthetic and experimental data.For the laser application, we presented a simple criterion that allows to distinguish between the thermal and non-thermal laser ablation regimes.

Nichtglatte Optimierungsprobleme in reflexiven Banachräumen treten in vielen Anwendungen auf. Häufig wird angenommen, dass alle vorkommenden Nichtdifferenzierbarkeiten durch Lipschitz-stetige Operatoren wie abs, min und max gegeben sind. Bei solchen Problemen kann es sich zum Beispiel um optimale Steuerungsprobleme mit möglicherweise nicht glatten Zielfunktionen handeln, welche durch partielle Differentialgleichungen (PDG) eingeschränkt sind, die ebenfalls nicht glatte Terme enthalten können. Eine effiziente und robuste Lösung erfordert eine Kombination numerischer Simulationen und spezifischer Optimierungsalgorithmen. Lokal Lipschitz-stetige, nichtglatte Nemytzkii-Operatoren, welche direkt in der Problemformulierung auftreten, spielen eine wesentliche Rolle in der Untersuchung der zugrundeliegenden Optimierungsprobleme. In dieser Dissertation werden zwei spezifische Methoden und Algorithmen zur Lösung solcher nichtglatter Optimierungsprobleme in reflexiven Banachräumen vorgestellt und diskutiert. Als erste Lösungsmethode wird in dieser Dissertation die Minimierung von nichtglatten Operatoren in reflexiven Banachräumen mittels sukzessiver quadratischer Überschätzung vorgestellt, SALMIN. Ein neuartiger Optimierungsansatz für Optimierungsprobleme mit nichtglatten elliptischen PDG-Beschränkungen, welcher auf expliziter Strukturausnutzung beruht, stellt die zweite Lösungsmethode dar, SCALi. Das zentrale Merkmal dieser Methoden ist ein geeigneter Umgang mit Nichtglattheiten. Besonderes Augenmerk liegt dabei auf der zugrundeliegenden nichtglatten Struktur des Problems und der effektiven Ausnutzung dieser, um das Optimierungsproblem auf angemessene und effiziente Weise zu lösen. / Non-smooth optimization problems in reflexive Banach spaces arise in many applications. Frequently, all non-differentiabilities involved are assumed to be given by Lipschitz-continuous operators such as abs, min and max. For example, such problems can refer to optimal control problems with possibly non-smooth objective functionals constrained by partial differential equations (PDEs) which can also include non-smooth terms. Their efficient as well as robust solution requires numerical simulations combined with specific optimization algorithms. Locally Lipschitz-continuous non-smooth non-linearities described by appropriate Nemytzkii operators which arise directly in the problem formulation play an essential role in the study of the underlying optimization problems. In this dissertation, two specific solution methods and algorithms to solve such non-smooth optimization problems in reflexive Banach spaces are proposed and discussed. The minimization of non-smooth operators in reflexive Banach spaces by means of successive quadratic overestimation is presented as the first solution method, SALMIN. A novel structure exploiting optimization approach for optimization problems with non-smooth elliptic PDE constraints constitutes the second solution method, SCALi. The central feature of these methods is the appropriate handling of non-differentiabilities. Special focus lies on the underlying structure of the problem stemming from the non-smoothness and how it can be effectively exploited to solve the optimization problem in an appropriate and efficient way.