Robust a priori and a posteriori error analysis for the approximation of Allen–Cahn and Ginzburg–Landau equations past topological changes

A priori and a posteriori error estimates are derived for the numerical approximation of scalar and complex valued phase field models. Particular attention is devoted to the dependence of the estimates on a small parameter and to the validity of the estimates in the presence of topological changes in the solution that represents singular points in the evolution. For typical singularities the estimates depend on the inverse of the parameter in a polynomial as opposed to exponential dependence of estimates resulting from a straightforward error analysis. The estimates naturally lead to adaptive mesh refinement and coarsening algorithms. Numerical experiments illustrate the reliability and efficiency of this approach for the evolution of interfaces and vortices that undergo topological changes

Uniform decrease of alpha-global field power induced by intermittent photic stimulation of healthy subjects

Nineteen-channel EEGs were recorded from the scalp surface of 30 healthy subjects (16 males and 14 females, mean age: 34 years, SD: 11.7 years) at rest and under trains of intermittent photic stimulation (IPS) at rates of 5, 10 and 20 Hz. Digitalized data were submitted to spectral analysis with fast fourier transformation providing the basis for the computation of global field power (GFP). For quantification, GFP values in the frequency ranges of 5, 10 and 20 Hz at rest were divided by the corresponding data obtained under IPS. All subjects showed a photic driving effect at each rate of stimulation. GFP data were normally distributed, whereas ratios from photic driving effect data showed no uniform behavior due to high interindividual variability. Suppression of alpha-power after IPS with 10 Hz was observed in about 70% of the volunteers. In contrast, ratios of alpha-power were unequivocal in all subjects: IPS at 20 Hz always led to a suppression of alpha-power. Dividing alpha-GFP with 20-Hz IPS by alpha-GFP at rest (R = a-GFPIPS/a-GFPrest) thus resulted in ratios lower than 1. We conclude that ratios from GFP data with 20-Hz IPS may provide a suitable paradigm for further investigations. Key words: EEG, Brain mapping, Intermittent photic stimulation, IPS, Global field power ratio

Transfer Function Optimization for Comparative Volume Rendering

Direct volume rendering is often used to compare different 3D scalar fields. The choice of the transfer function which maps scalar values to color and opacity plays a critical role in this task. We present a technique for the automatic optimization of a transfer function so that rendered images of a second field match as good as possible images of a field that has been rendered with some other transfer function. This enables users to see whether differences in the visualizations can be solely attributed to the choice of transfer function or remain after optimization. We propose and compare two different approaches to solve this problem, a voxel-based solution solving a least squares problem, and an image-based solution using differentiable volume rendering for optimization. We further propose a residual-based visualization to emphasize the differences in information content