Stochastic Quasi-Fej\'er Block-Coordinate Fixed Point Iterations with Random Sweeping

This work proposes block-coordinate fixed point algorithms with applications to nonlinear analysis and optimization in Hilbert spaces. The asymptotic analysis relies on a notion of stochastic quasi-Fej\'er monotonicity, which is thoroughly investigated. The iterative methods under consideration feature random sweeping rules to select arbitrarily the blocks of variables that are activated over the course of the iterations and they allow for stochastic errors in the evaluation of the operators. Algorithms using quasinonexpansive operators or compositions of averaged nonexpansive operators are constructed, and weak and strong convergence results are established for the sequences they generate. As a by-product, novel block-coordinate operator splitting methods are obtained for solving structured monotone inclusion and convex minimization problems. In particular, the proposed framework leads to random block-coordinate versions of the Douglas-Rachford and forward-backward algorithms and of some of their variants. In the standard case of $m=1$ block, our results remain new as they incorporate stochastic perturbations

The Geometry of Monotone Operator Splitting Methods

We propose a geometric framework to describe and analyze a wide array of operator splitting methods for solving monotone inclusion problems. The initial inclusion problem, which typically involves several operators combined through monotonicity-preserving operations, is seldom solvable in its original form. We embed it in an auxiliary space, where it is associated with a surrogate monotone inclusion problem with a more tractable structure and which allows for easy recovery of solutions to the initial problem. The surrogate problem is solved by successive projections onto half-spaces containing its solution set. The outer approximation half-spaces are constructed by using the individual operators present in the model separately. This geometric framework is shown to encompass traditional methods as well as state-of-the-art asynchronous block-iterative algorithms, and its flexible structure provides a pattern to design new ones

The Glowinski–Le Tallec splitting method revisited in the framework of equilibrium problems in Hilbert spaces

In this paper, we introduce a new approach for solving equilibrium problems in Hilbert spaces. First, we transform the equilibrium problem into the problem of finding a zero of a sum of two maximal monotone operators. Then, we solve the resulting problem using the Glowinski–Le Tallec splitting method and we obtain a linear rate of convergence depending on two parameters. In particular, we enlarge significantly the range of these parameters given rise to the convergence. We prove that the sequence generated by the new method converges to a global solution of the considered equilibrium problem. Finally, numerical tests are displayed to show the efficiency of the new approach

Fast, Adaptive Algorithms for Flow Problems

Time-accurate simulations of physical phenomena (e.g., ocean dynamics, weather, and combustion) are essential to economic development and the well-being of humanity. For example, the economic toll hurricanes wrought on the United States in 2017 exceeded $\$200$ billon dollars. To mitigate the damage, the accurate and timely forecasting of hurricane paths are essential. Ensemble simulations, used to calculate mean paths via multiple realizations, are an invaluable tool in estimating uncertainty, understanding rare events, and improving forecasting. The main challenge in the simulation of fluid flow is the complexity (runtime, memory requirements, and efficiency) of each realization. This work confronts each of these challenges with several novel ensemble algorithms that allow for the fast, efficient computation of flow problems, all while reducing memory requirements. The schemes in question exploit the saddle-point structure of the incompressible Navier-Stokes (NSE) and Boussinesq equations by relaxing incompressibility appropriately via artificial compressibility (AC), yielding algorithms that require far fewer resources to solve while retaining time-accuracy. Paired with an implicit-explicit (IMEX) ensemble method that employs a shared coefficient matrix, we develop, analyze, and validate novel schemes that reduce runtime and memory requirements. Using these methods as building blocks, we then consider schemes that are time-adaptive, i.e., schemes that utilize varying timestep sizes. The consideration of time-adaptive artficial compressibility methods, used in the algorithms mentioned above, also leads to the study of a new slightly-compressible fluid flow continuum model. This work demonstrates stability and weak convergence of the model to the incompressible NSE, and examines two associated time-adaptive AC methods. We show that these methods are unconditionally, nonlinearly, long-time stable and demonstrate numerically their accuracy and efficiency. The methods described above are designed for laminar flow; turbulent flow is addressed with the introduction of a novel one-equation unsteady Reynolds-averaged Navier-Stokes (URANS) model with multiple improvements over the original model of Prandtl. This work demonstrates analytically and numerically the advantages of the model over the original

Novel Image Processing Methods for Improved Fetal Brain MRI

Fetal magnetic resonance imaging (MRI) has been increasingly used as a powerful complement imaging modality to ultrasound imaging (US) for the clinical evaluation of prenatal abnormalities. Specifically, clinical application of fetal MRI has been significantly improved in the nineties by hardware and software advances with the development of ultrafast multi-slice T2-weighted (T2w) acquisition sequences able to freeze the unpredictable fetal motion and provide excellent soft-tissue contrast. Fetal motion is indeed the major challenge in fetal MRI and slice acquisition time should be kept as short as possible. As a result, typical fetal MRI examination involves the acquisition of a set of orthogonally planned scans of thick two-dimensional slices, largely free of intra-slice motion artifacts. The poor resolution in the slice-select dimension as well as possible motion occurring between slices limits further quantitative data analysis, which is the key for a better understanding of the developing brain but also the key for the determination of operator-independent biomarkers that might significantly facilitate fetal diagnosis and prognosis. To this end, several research groups have developed in the past ten years advanced image processing methods, often denoted by motion-robust super-resolution (SR) techniques, to reconstruct from a set of clinical low-resolution (LR) scans, a high-resolution (HR) motion-free volume. SR problem is usually modeled as a linear inverse problem describing the imaging degradation due to acquisition and fetal motion. Typically, such approaches consist in iterating between slice motion estimation that estimates the motion parameters and SR that recovers the HR image given the estimated degradation model. This thesis focuses on the development of novel advanced image processing methods, which have enabled the design of a completely automated reconstruction pipeline for fetal MRI. The proposed techniques help in improving state-of-the-art fetal MRI reconstruction in terms of efficiency, robustness and minimized user-interactions, with the ultimate goal of being translated to the clinical environment. The first part focuses on the development of a more efficient Total Variation (TV)-regularized optimization algorithm for the SR problem. The algorithm uses recent advances in convex optimization with a novel adaptive regularization strategy to offer simultaneously fast, accurate and robust solutions to the fetal image recovery problem. Extensive validations on both simulated fetal and real clinical data show the proposed algorithm is highly robust in front of motion artifacts and that it offers the best trade-off between speed and accuracy for fetal MRI recovery as in comparison with state-of-the art methods. The second part focuses on the development of a novel automatic brain localization and extraction approach based on template-to-slice block matching and deformable slice-totemplate registration. Asmost fetal brain MRI reconstruction algorithms rely only on brain tissue-relevant voxels of low-resolution (LR) images to enhance the quality of inter-slice motion correction and image reconstruction, the fetal brain needs to be localized and extracted as a first step. These tasks generally necessitate user interaction, manually or semi-automatically done. Our methods have enabled the design of completely automated reconstruction pipeline that involves intensity normalization, inter-slice motion estimation, and super-resolution. Quantitative evaluation on clinical MRI scans shows that our approach produces brain masks that are very close to manually drawn brain masks, and ratings performed by two expert observers show that the proposed pipeline achieves similar reconstruction quality to reference reconstruction based on manual slice-by-slice brain extraction without any further effort. The third part investigates the possibility of automatic cortical folding quantification, one of the best biomarkers of brain maturation, by combining our automatic reconstruction pipeline with a state-of-the-art fetal brain tissue segmentation method and existing automated tools provided for adult brain’s cortical folding quantification. Results indicate that our reconstruction pipeline can provide HR MR images with sufficient quality that enable the use of surface tessellation and active surface algorithms similar to those developed for adults to extract meaningful information about fetal brain maturation. Finally, the last part presents new methodological improvements of the reconstruction pipeline aiming at improving the quality of the image for quantitative data analysis, whose accuracy is highly dependent on the quality and resolution of the reconstructed image. In particular, it presents a more consistent and global magnetic bias field correction method which takes advantage of the super-resolution framework to provide a final reconstructed image quasi free of the smooth bias field. Then, it presents a new TV SR algorithm that uses the Huber norm in the data fidelity term to be more robust to non-Gaussian outliers. It also presents the design of a novel joint reconstruction-segmentation framework and the development of a novel TV SR algorithm driven by segmentation to produce images with enhanced edge information that could ultimately improve their segmentation. Finally, it preliminary investigates the capability of increasing the resolution in the in-plane dimensions using SR to ultimately reduce the partial volume effect

Geometrical Modeling and Numerical Simulation of Heterogeneous Materials

The discretization of the considered body or material by finite elements is a crucial part of the Finite Element Method in addition to the material modeling and the element formulation. Thereby, the treatment of heterogeneous materials is an advanced challenge, because the interfaces between the individual constituents must be taken into account in addition to the free surfaces. In many materials these interfaces exhibit complex geometries, since they are built up in growth and transformation processes. In the present work the numerical analysis of such materials is presented starting from the geometrical construction and ending up with the evaluation of the computational results. The first part is concerned with the simulation of diseased blood vessels and focusses on the reconstruction of patient-specific arterial geometries. The results of two- and three-dimensional finite element simulations show the field of application of the presented method. The consideration of the heterogeneity of modern two-phase steels in the numerical simulation is given in the following part of the present work. Therein the focus is on the application of geometrically simplified structures, which exhibit a similar mechanical response compared to the real microstructure. The applicability of the proposed method is shown in different boundary value problems using a direct micro-macro transition approach.Bei Simulationen unter Verwendung der Finite-Elemente-Methode spielt neben der Materialmodellierung und Elementformulierung die Diskretisierung des zu untersuchenden Körpers oder Materials durch finite Elemente eine große Rolle. Diese Aufgabe wird erschwert, wenn es sich um heterogene Materialien handelt. Bei diesen müssen zusätzlich zu den äußeren freien Oberflächen die inneren Grenzschichten zwischen den jeweiligen Individuen berücksichtigt werden. In vielen Materialien sind diese Grenzflächen durch Wachstums- oder Umwandlungsprozesse entstanden, können somit auch komplexe Strukturen aufweisen und erschweren die geometrische Beschreibung. Die vorliegende Arbeit beschäftigt sich im Wesentlichen mit der numerischen Analyse solcher Materialien ausgehend von der Konstruktion der Geometrien bis hin zur Auswertung der Simulationsergebnisse. Der erste Teil der Arbeit beschäftigt sich mit der Simulation von erkrankten Blutgefäßen und geht dort vor allem auf die Rekonstruktion von patienten-spezifischen Arteriengeometrien ein. Die Ergebnisse von zwei- und dreidimensionalen FE-Berechnung verdeutlichen das Einsatzgebiet der vorgestellten Methodik. Die Berücksichtigung der Heterogenität moderner Zweiphasenstähle in der numerischen Simulation wird im anschließenden Teil der Arbeit vorgestellt. Hierbei liegt der Schwerpunkt auf dem Einsatz von geometrisch vereinfachten Ersatzstrukturen, die ein vergleichbares mechanisches Antwortverhalten zur realen Mikrostruktur liefern. Die Anwendbarkeit dieser Methode wird in verschiedenen Randwertproblemen unter Einsatz eines direkten Mikro-Makro Übergangs gezeigt

Proceedings of the 2018 Canadian Society for Mechanical Engineering (CSME) International Congress

Published proceedings of the 2018 Canadian Society for Mechanical Engineering (CSME) International Congress, hosted by York University, 27-30 May 2018