Modern Regularization Methods for Inverse Problems

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift towards modern nonlinear regularization methods, including their analysis, applications and issues for future research. In particular we will discuss variational methods and techniques derived from them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.Leverhulme Trust Early Career Fellowship ‘Learning from mistakes: a supervised feedback-loop for imaging applications’ Isaac Newton Trust Cantab Capital Institute for the Mathematics of Information ERC Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse German Ministry for Science and Education (BMBF) project MED4D EPSRC grant EP/K032208/

Techniques d'optimisation non lisse avec des applications en automatique et en mécanique des contacts

L'optimisation non lisse est une branche active de programmation non linéaire moderne, où l'objectif et les contraintes sont des fonctions continues mais pas nécessairement différentiables. Les sous-gradients généralisés sont disponibles comme un substitut à l'information dérivée manquante, et sont utilisés dans le cadre des algorithmes de descente pour se rapprocher des solutions optimales locales. Sous des hypothèses réalistes en pratique, nous prouvons des certificats de convergence vers les points optimums locaux ou critiques à partir d'un point de départ arbitraire. Dans cette thèse, nous développons plus particulièrement des techniques d'optimisation non lisse de type faisceaux, où le défi consiste à prouver des certificats de convergence sans hypothèse de convexité. Des résultats satisfaisants sont obtenus pour les deux classes importantes de fonctions non lisses dans des applications, fonctions C1-inférieurement et C1-supérieurement. Nos méthodes sont appliquées à des problèmes de design dans la théorie du système de contrôle et dans la mécanique de contact unilatéral et en particulier, dans les essais mécaniques destructifs pour la délaminage des matériaux composites. Nous montrons comment ces domaines conduisent à des problèmes d'optimisation non lisse typiques, et nous développons des algorithmes de faisceaux appropriés pour traiter ces problèmes avec succèsNonsmooth optimization is an active branch of modern nonlinear programming, where objective and constraints are continuous but not necessarily differentiable functions. Generalized subgradients are available as a substitute for the missing derivative information, and are used within the framework of descent algorithms to approximate local optimal solutions. Under practically realistic hypotheses we prove convergence certificates to local optima or critical points from an arbitrary starting point. In this thesis we develop especially nonsmooth optimization techniques of bundle type, where the challenge is to prove convergence certificates without convexity hypotheses. Satisfactory results are obtained for two important classes of nonsmooth functions in applications, lower- and upper-C1 functions. Our methods are applied to design problems in control system theory and in unilateral contact mechanics and in particular, in destructive mechanical testing for delamination of composite materials. We show how these fields lead to typical nonsmooth optimization problems, and we develop bundle algorithms suited to address these problems successfully

Higher-Order DGFEM Transport Calculations on Polytope Meshes for Massively-Parallel Architectures

In this dissertation, we develop improvements to the discrete ordinates (S_N) neutron transport equation using a Discontinuous Galerkin Finite Element Method (DGFEM) spatial discretization on arbitrary polytope (polygonal and polyhedral) grids compatible for massively-parallel computer architectures. Polytope meshes are attractive for multiple reasons, including their use in other physics communities and their ease in handling local mesh refinement strategies. In this work, we focus on two topical areas of research. First, we discuss higher-order basis functions compatible to solve the DGFEM S_N transport equation on arbitrary polygonal meshes. Second, we assess Diffusion Synthetic Acceleration (DSA) schemes compatible with polytope grids for massively-parallel transport problems. We first utilize basis functions compatible with arbitrary polygonal grids for the DGFEM transport equation. We analyze four different basis functions that have linear completeness on polygons: the Wachspress rational functions, the PWL functions, the mean value coordinates, and the maximum entropy coordinates. We then describe the procedure to extend these polygonal linear basis functions into the quadratic serendipity space of functions. These quadratic basis functions can exactly interpolate monomial functions up to order 2. Both the linear and quadratic sets of basis functions preserve transport solutions in the thick diffusion limit. Maximum convergence rates of 2 and 3 are observed for regular transport solutions for the linear and quadratic basis functions, respectively. For problems that are limited by the regularity of the transport solution, convergence rates of 3/2 (when the solution is continuous) and 1/2 (when the solution is discontinuous) are observed. Spatial Adaptive Mesh Refinement (AMR) achieved superior convergence rates than uniform refinement, even for problems bounded by the solution regularity. We demonstrated accuracy in the AMR solutions by allowing them to reach a level where the ray effects of the angular discretization are realized. Next, we analyzed DSA schemes to accelerate both the within-group iterations as well as the thermal upscattering iterations for multigroup transport problems. Accelerating the thermal upscattering iterations is important for materials (e.g., graphite) with significant thermal energy scattering and minimal absorption. All of the acceleration schemes analyzed use a DGFEM discretization of the diffusion equation that is compatible with arbitrary polytope meshes: the Modified Interior Penalty Method (MIP). MIP uses the same DGFEM discretization as the transport equation. The MIP form is Symmetric Positive De_nite (SPD) and e_ciently solved with Preconditioned Conjugate Gradient (PCG) with Algebraic MultiGrid (AMG) preconditioning. The analysis from previous work was extended to show MIP's stability and robustness for accelerating 3D transport problems. MIP DSA preconditioning was implemented in the Parallel Deterministic Transport (PDT) code at Texas A&M University and linked with the HYPRE suite of linear solvers. Good scalability was numerically verified out to around 131K processors. The fraction of time spent performing DSA operations was small for problems with sufficient work performed in the transport sweep (O(10^3) angular directions). Finally, we have developed a novel methodology to accelerate transport problems dominated by thermal neutron upscattering. Compared to historical upscatter acceleration methods, our method is parallelizable and amenable to massively parallel transport calculations. Speedup factors of about 3-4 were observed with our new method

Optimisation of postbuckling stiffened composite structures

The thesis starts off with an introductory chapter on composite materials. This includes a definition of composites, a brief history of composite materials, their use in aerostructures (primarily as stiffened structures), and also optimization of composite structures. A literature review is then presented on postbuckling stiffened structures. This includes both experimental investigations on stiffened composite panels and investigations into secondary instabilities and mode jumping as well as their numerical modelling. Next, the Finite Element (FE) modelling of posthuckling stiffened structures is discussed, relating how ABAQUS models are set up in order to trace stiffened composite panels' buckling and postbuckling responses. An experimental programme conducted on an I-stiffened panel is described, where the panel was tested in compression until collapse. The buckling and postbuckling characteristics of the panel are presented, and then an FE model is described together with its predicted numerical behaviour of the panel's buckling and postbuckling characteristics. Focus then shifts to the modelling of failure in composites, in particular delamination failure. A literature review is conducted, looking at the use of both the Virtual Crack Closure Technique (VCCT) and interface elements in delamination modelling. Two stiffener runout models, representing two specimens previously tested experimentally, are then developed to illustrate how interface elements may be used to model mixed mode delamination. The previously discussed panel is revisited, and a global-local modelling approach used to model the skin-stiffener interface. FE models of a stiffened cylindrical shell are also considered, and again the postbuckling characteristics of the shell are compared with experimental results. . The thesis then moves on to optimization of composite structures. This starts off with a literature review of existing optimization methodologies. A Genetic Algorithm (GA) is devised to increase the damage resistance of the I-stiffened panel. The global-local ABAQUS model discussed earlier is used in conjunction with the GA in order to find a revised stacking sequence of both the panel flanges and skin so as to minimize skin-stiffener debonding subject to a variety of design constraints. A second optimization is then presented, this time linked to the FE model of the stiffened cylindrical shell. The objective is to increase the collapse load of the shell, again subject to specific design constraints. The thesis concludes by summarising the importance of the work conducted. FE models were created and validated against experimental work in order to model a variety of composite stiffened structures in their buckling and postbuckling regimes. These models were able to capture the failure characteristics of these structures relating to delamination at the skin-stiffener interface, a phenomenon widely observed experimentally. Various optimizations, able to account for failure mechanisms which may occur prior to overall structural collapse, were then conducted on the analysed structures in order to obtain more damage resistant designs.Imperial Users onl