A Parallel Geometric Multigrid Method for Adaptive Finite Elements

Applications in a variety of scientific disciplines use systems of Partial Differential Equations (PDEs) to model physical phenomena. Numerical solutions to these models are often found using the Finite Element Method (FEM), where the problem is discretized and the solution of a large linear system is required, containing millions or even billions of unknowns. Often times, the domain of these solves will contain localized features that require very high resolution of the underlying finite element mesh to accurately solve, while a mesh with uniform resolution would require far too much computational time and memory overhead to be feasible on a modern machine. Therefore, techniques like adaptive mesh refinement, where one increases the resolution of the mesh only where it is necessary, must be used. Even with adaptive mesh refinement, these systems can still be on the order of much more than a million unknowns (large mantle convection applications like the ones in [90] show simulations on over 600 billion unknowns), and attempting to solve on a single processing unit is infeasible due to limited computational time and memory required. For this reason, any application code aimed at solving large problems must be built using a parallel framework, allowing the concurrent use of multiple processing units to solve a single problem, and the code must exhibit efficient scaling to large amounts of processing units. Multigrid methods are currently the only known optimal solvers for linear systems arising from discretizations of elliptic boundary valued problems. These methods can be represented as an iterative scheme with contraction number less than one, independent of the resolution of the discretization [24, 54, 25, 103], with optimal complexity in the number of unknowns in the system [29]. Geometric multigrid (GMG) methods, where the hierarchy of spaces are defined by linear systems of finite element discretizations on meshes of decreasing resolution, have been shown to be robust for many different problem formulations, giving mesh independent convergence for highly adaptive meshes [26, 61, 83, 18], but these methods require specific implementations for each type of equation, boundary condition, mesh, etc., required by the specific application. The implementation in a massively parallel environment is not obvious, and research into this topic is far from exhaustive. We present an implementation of a massively parallel, adaptive geometric multigrid (GMG) method used in the open-source finite element library deal.II [5], and perform extensive tests showing scaling of the v-cycle application on systems with up to 137 billion unknowns run on up to 65,536 processors, and demonstrating low communication overhead of the algorithms proposed. We then show the flexibility of the GMG by applying the method to four different PDE systems: the Poisson equation, linear elasticity, advection-diffusion, and the Stokes equations. For the Stokes equations, we implement a fully matrix-free, adaptive, GMG-based solver in the mantle convection code ASPECT [13], and give a comparison to the current matrix-based method used. We show improvements in robustness, parallel scaling, and memory consumption for simulations with up to 27 billion unknowns and 114,688 processors. Finally, we test the performance of IDR(s) methods compared to the FGMRES method currently used in ASPECT, showing the effects of the flexible preconditioning used for the Stokes solves in ASPECT, and the demonstrating the possible reduction in memory consumption for IDR(s) and the potential for solving large scale problems. Parts of the work in this thesis has been submitted to peer reviewed journals in the form of two publications ([36] and [34]), and the implementations discussed have been integrated into two open-source codes, deal.II and ASPECT. From the contributions to deal.II, including a full length tutorial program, Step-63 [35], the author is listed as a contributing author to the newest deal.II release (see [5]). The implementation into ASPECT is based on work from the author and Timo Heister. The goal for the work here is to enable the community of geoscientists using ASPECT to solve larger problems than currently possible. Over the course of this thesis, the author was partially funded by the NSF Award OAC-1835452 and by the Computational Infrastructure in Geodynamics initiative (CIG), through the NSF under Award EAR-0949446 and EAR-1550901 and The University of California -- Davis

Breast density classification using deep learning

Breast density screenings are an accepted means to determine a patient\u27s predisposed risk of breast cancer development. Although the direct correlation is not fully understood, breast cancer risk increases with higher levels of mammographic breast density. Radiologists visually assess a patient\u27s breast density using mammogram images and assign a density score based on four breast density categories outlined by the Breast Imaging and Reporting Data Systems (BI-RADS). There have been efforts to develop automated tools that assist radiologists with increasing workloads and to help reduce the intra- and inter-rater variability between radiologists. In this thesis, I explored two deep-learning-based approaches on breast density classification. First, I developed and experimented with algorithms using deep learning models (such as Inception V3 and ViT) to classify patients according to BI-RADS using various types of digital mammograms. Second, with the need to provide not only such classification but also a quantitative measure of breast density to help standardize assessments across radiologists, I applied a deep learning based semantic segmentation model, DeepLabV3, to predict density percentages which then were used to provide a linear and probability scale

Voter experience of corrupt officials is an overlooked reason for the electoral success of radical right parties

The radical right has had a good few years, with the economic crisis providing fertile ground for a political discourse centred around immigration and economic injustice. Conrad Ziller and Thomas Schübel share research which shows that one of the key, though overlooked, drivers of the rise of parties like the Finns Party in Finland and the Freedom Party of Austria is voter experience of corruption by officials

Environmental Freshwater Resources and Conflict: Case Studies From the Former Soviet Union

Natural resources have become a factor of interest in conflict. Unfortunately scholars and policymakers alike choose to focus on oil and natural gas, not freshwater, even when freshwater resources play a significant role in their conflict of interest. Utilizing a theoretical logic developed from Jane Dawson\u27s resource-identity model of conflict, freshwater resources are understood to be fundamental factors in the causation of conflicts as well as in the strategy and tactics used within the conflict. Case studies from Central Asia, Georgia-Abkhazia, and Russian intervention demonstrate this in the Former Soviet Union

Transitions : individuelle Handhabung und Verarbeitungsformen institutionellen Wandels

Unter Transitionen werden hier allgemein unstete, diskontinuierliche Übergangsprozesse verstanden. In der Selbstbeobachtung erscheinen sie z.B. als Brüche, überraschende Ereignisse, ungeahnte Chancen oder nie für möglich gehaltene Schocks. Retrospektiv jedenfalls – positiv wie negativ bewertet – als entscheidende Weichenstellungen, die später nachfolgende Entscheidungen in einschneidendem Umfang wenn schon nicht determinieren, so jedenfalls aber nachhaltig oder dauerhaft prägen. Als unstet werden sie deswegen eingeschätzt, weil Akteure heute davon zunehmend „unvorhergesehen“ und nicht planbar betroffen sind

Explaining Inflation Persistence by a Time-Varying Taylor Rule

In a simple New Keynesian model, we derive a closed form solution for the inflation persistence parameter as a function of the policy weights in the central bank’s Taylor rule. By estimating the time-varying weights that the FED attaches to inflation and the output gap, we show that the empirically observed changes in U.S. inflation persistence during the period 1975 to 2010 can be well explained by changes in the conduct of monetary policy. Our findings are in line with Benati’s (2008) view that inflation persistence should not be considered a structural parameter in the sense of Lucas

Explaining Inflation-Gap Persistence by a Time-Varying Taylor Rule

In a simple New Keynesian model, we derive a closed form solution for the inflation-gap persistence parameter as a function of the policy weights in the central bank’s Taylor rule. By estimating the time-varying weights that the FED attaches to inflation and the output gap, we show that the empirically observed changes in U.S. inflation-gap persistence during the period 1975 to 2010 can be well explained by changes in the conduct of monetary policy. Our findings are in line with Benati’s (2008) view that inflation persistence should not be considered a structural parameter in the sense of Lucas