Crystal Structures of the [2.2.2]Cryptates [Cd(cryptand 222)][CdCl4] and [Hg(cryptand 222)][Hg2Cl6] with 8-Coordinated Metal Ions

Reactions of [2.2.2]cryptand, 4,7,13,16,21,24-Hexaoxa-1,10-diazabicyclo-[8.8.8]hexacosane, (“cryptand 222”) with cadmium chloride and mercuric chloride yielded crystals of [Cd(cryptand 222)][CdCl4] (1) and [Hg(cryptand 222)][Hg2Cl6] (2). 1 crystallizes tetragonally, space group P42/n, Z = 4, a = 1239.8(2), c = 1801.2(8) pm; 2 is monoclinic, space group P2/n, Z = 2, a = 1060.3(3), b = 955.2(2), c = 1607.0(3) pm. and β = 108.85(2)°. In the cations of both complexes the metal ions are enclosed in the bicyclic ligand and show 8-coordination to its six oxygen and two nitrogen atoms, the coordination polyhedron is a distorted cube for 1 and a bicapped trigonal antiprism for 2. The anion of 2 is dimeric, the two mercury atoms are connected via chlorine bridges with Hg-Cl distances of 258 and 278 pm, the Hg-Cl bond lengths to the two terminal chloro ligands being shorter at 236 and 238 pm

Parallel adaptive FETI-DP using lightweight asynchronous dynamic load balancing

A parallel FETI-DP domain decomposition method using an adaptive coarse space is presented. The implementation builds on a recently introduced adaptive FETI-DP approach for elliptic problems in three dimensions and uses small, local eigenvalue problems for faces and, additionally, for a small number of edges. The condition number of the preconditioned operator then satisfies a bound which is independent of coefficient heterogeneities in the problem. The computational cost of the local eigenvalue problems is not negligible, and also a significant load imbalance can be introduced. As a remedy, certain eigenvalue problems are discarded by a theory-guided heuristic strategy, based on the diagonal entries of the stiffness matrices. Additionally, a lightweight pairwise dynamic load balancing strategy is implemented for the eigenvalue problems. The load balancing is supervised by an orchestrating rank using asynchronous point-to-point communication. The resulting method shows good weak and strong scalability up to thousands of cores while fast convergence is obtained even for heterogeneous problems

Adaptive FETI-DP and BDDC methods for highly heterogeneous elliptic finite element problems in three dimensions

Numerical methods are often well-suited for the solution of (elliptic) partial differential equations (PDEs) modeling naturally occuring processes. Many different solvers can be applied to systems which are obtained after discretization by the finite element method. Parallel architectures in modern computers facilitate the efficient use of diverse divide and conquer strategies. The intuitive approach, to divide a large (global) problem into subproblems, which are then solved in parallel, can significantly reduce the solution time. It is obvious that the solvers on the local subproblems then should deliver the contributions of the global solution restricted to the subdomains of computational region. The class of domain decomposition methods provides widely-used iterative algorithms for the parallel solution of implicit finite element problems. Often, an additional coarse space, which introduces a coupling between the subdomains, is used to ensure a global transport of information between the subdomains across the entire domain. The FETI-DP and BDDC domain decomposition methods are highly scalable parallel algorithms. However, when the parameter or coefficient distribution in the underlying partial differential equation becomes highly heterogeneous, classical methods, with a priori chosen coarse spaces, might not converge in a limited number of iterations. A remedy is offered by problem-dependent coarse spaces. These coarse spaces can be provided by adaptive methods, which then can improve the convergence at the cost of additional constraints. In this thesis, we introduce robust FETI-DP and BDDC methods for three-dimensional problems. These methods incorporate constraints, which are computed from local eigenvalue problems on faces and edges between subdomains, into the coarse space. The implementation of the constraints is performed by a deflation or balancing approach or by partial finite element assembly after a transformation of basis. For the latter, we introduce the generalized transformation-of-basis approach and show its correspondence to a deflation or balancing approach. An efficient parallel implementation of adaptive FETI-DP is discussed in the last part of this thesis. We provide weak and strong parallel scalability results for our adaptive algorithm executed on the supercomputer magnitUDE of the University of Duisburg-Essen. For weak scaling, we can show very good results up to 4,096 cores. We can also present very good strong scaling results up to 864 cores

Mathematical modeling of infectious diseases: An introduction

Non-pharmaceutical interventions (NPIs) are important to mitigate the spread of infectious diseases as long as no vaccination or outstanding medical treatments are available. We assess the effectiveness of the sets of non-pharmaceutical interventions that were in place during the course of the Coronavirus disease 2019 (Covid-19) pandemic in Germany. Our results are based on hybrid models, combining recent SIR-type models on local scales with spatial resolution. In order to account for the age-dependence of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), we include realistic prepandemic and recently recorded contact patterns between age groups. The implementation of non-pharmaceutical interventions will occur on changed contact patterns, improved isolation, or reduced infectiousness when, e.g., wearing masks. In order to account for spatial heterogeneity, we use a graph approach and we include high-quality information on commuting activities combined with traveling information from social networks. The remaining uncertainty will be accounted for by a large number of randomized simulation runs. Based on the derived factors for the effectiveness of different non-pharmaceutical interventions over the past months, we provide different forecast scenarios for the upcoming time

Reinforcement magnitudes modulate subthalamic beta band activity in patients with Parkinson's disease

We set out to investigate whether beta oscillations in the human basal ganglia are modulated during reinforcement learning. Based on previous research, we assumed that beta activity might either reflect the magnitudes of individuals' received reinforcements (reinforcement hypothesis), their reinforcement prediction errors (dopamine hypothesis) or their tendencies to repeat versus adapt responses based upon reinforcements (status-quo hypothesis). We tested these hypotheses by recording local field potentials (LFPs) from the subthalamic nuclei of 19 Parkinson's disease patients engaged in a reinforcement-learning paradigm. We then correlated patients' reinforcement magnitudes, reinforcement prediction errors and response repetition tendencies with task-related power changes in their LFP oscillations. During feedback presentation, activity in the frequency range of 14 to 27 Hz (beta spectrum) correlated positively with reinforcement magnitudes. During responding, alpha and low beta activity (6 to 18 Hz) was negatively correlated with previous reinforcement magnitudes. Reinforcement prediction errors and response repetition tendencies did not correlate significantly with LFP oscillations. These results suggest that alpha and beta oscillations during reinforcement learning reflect patients' observed reinforcement magnitudes, rather than their reinforcement prediction errors or their tendencies to repeat versus adapt their responses, arguing both against an involvement of phasic dopamine and against applicability of the status-quo theory

Local spectra of adaptive domain decomposition methods

We compare the spectra of local generalized eigenvalue problems in different adaptive coarse spaces for overlapping and nonoverlapping domain decomposition methods. In particular, we compare the AGDSW (Adaptive Generalized Dryja-Smith-Widlund), the OS-ACMS (Overlapping Schwarz-Approximate Component Mode Synthesis), and the SHEM (Spectral Harmonically Enriched Multiscale) coarse spaces for overlapping Schwarz methods, the GenEO (Generalized Eigenproblems in the Overlaps) coarse space for FETI-1 and BDD methods, and two approaches based on estimates for the $P_D$ operator for FETI-DP and BDDC methods. Therefore, we consider eight different two-dimensional coefficient functions with jumps ranging from simple channels to a realistic microstructure of a dual-phase steel. We observe significant differences in the width of the gap between good and bad eigenvalues depending on the coefficient distribution. In addition to that, eigenvalue problems involving sophisticated but more expensive harmonic extensions or deluxe-scaling can reduce the number of bad eigenvalues