Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
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Simulation of the mode dynamics in broad-ridge laser diodes
Get PDFIn this publication a method to simulate the mode dynamics in broad-ridge laser diode is presented. These devices exhibit rich lateral mode dynamics in addition to longitudinal mode dynamics observed in narrow-ridge laser diodes. The mode dynamics are strongly influenced by higher order effects, which are described by effective interaction terms and can derived from the band structure and the carrier scattering in the quantum well. The spatial dependency of pump current densities plays a crucial role in lateral mode dynamics, and thus, a Drift-Diffusion model is employed to calculate the current densities with an additional capturing term
Prethermalization in aperiodically kicked many-body dynamics
Get PDFDriven many-body systems typically experience heating due to the lack of energy conservation. Heating may be suppressed for time-periodic drives, but little is known for less regular drive protocols. In this work, we investigate the heating dynamics in aperiodically kicked systems, specifically those driven by quasi-periodic Thue-Morse or a family of random sequences with -multipolar temporal correlations. We demonstrate that multiple heating channels can be eliminated even away from the high-frequency regime. The number of eliminated channels increases with multipolar order . We illustrate this in a classical kicked rotor chain where we find a long-lived prethermal regime. When the static Hamiltonian only involves the kinetic energy, the prethermal lifetime can strongly depend on the temporal correlations of the drive, with a power-law dependence on the kick strength , for which we can account using a simple linearization argument
Analysing Cerebrospinal Fluid with Explainable Deep Learning: From Diagnostics to Insights
No full textAim Analysis of cerebrospinal fluid (CSF) is essential for diagnostic workup of patients with neurological diseases and includes differential cell typing. The current gold standard is based on microscopic examination by specialised technicians and neuropathologists, which is time-consuming, labour-intensive and subjective. Methods We, therefore, developed an image analysis approach based on expert annotations of 123,181 digitised CSF objects from 78 patients corresponding to 15 clinically relevant categories and trained a multiclass convolutional neural network (CNN). Results The CNN classified the 15 categories with high accuracy (mean AUC 97.3%). By using explainable artificial intelligence (XAI), we demonstrate that the CNN identified meaningful cellular substructures in CSF cells recapitulating human pattern recognition. Based on the evaluation of 511 cells selected from 12 different CSF samples, we validated the CNN by comparing it with seven board-certified neuropathologists blinded for clinical information. Inter-rater agreement between the CNN and the ground truth was non-inferior (Krippendorff's alpha 0.79) compared with the agreement of seven human raters and the ground truth (mean Krippendorff's alpha 0.72, range 0.56-0.81). The CNN assigned the correct diagnostic label (inflammatory, haemorrhagic or neoplastic) in 10 out of 11 clinical samples, compared with 7-11 out of 11 by human raters. Conclusions Our approach provides the basis to overcome current limitations in automated cell classification for routine diagnostics and demonstrates how a visual explanation framework can connect machine decision-making with cell properties and thus provide a novel versatile and quantitative method for investigating CSF manifestations of various neurological diseases
Adaptive Grids in the Context of Algebraic Stabilizations for Convection-Diffusion-Reaction Equations
No full textThree algebraically stabilized finite element schemes for discretizing convection-diffusion-reaction equations are studied on adaptively refined grids. These schemes are the algebraic flux correction (AFC) scheme with Kuzmin limiter, the AFC scheme with BJK limiter, and the recently proposed Monotone Upwind-type Algebraically Stabilized (MUAS) method. Both, conforming closure of the refined grids and grids with hanging vertices are considered. A non-standard algorithmic step becomes necessary before these schemes can be applied on grids with hanging vertices. The assessment of the schemes is performed with respect to the satisfaction of the global discrete maximum principle (DMP), the accuracy, e.g., smearing of layers, and the efficiency in solving the corresponding nonlinear problems
Quenched homogenization of infinite range random conductance model on stationary point processes
Get PDFWe prove homogenization for elliptic long-range operators in the random conductance model on random stationary point processes in d dimensions with Dirichlet boundary conditions and with a jointly stationary coefficient field. Doing so, we identify 4 conditions on the point process and the coefficient field that have to be fulfilled at different stages of the proof in order to pass to the homogenization limit. The conditions can be clearly attributed to concentration of support, Rellich--Poincaré inequality, non-degeneracy of the homogenized matrix and ergodicity of the elliptic operator
Accurate evaluation of self-heterodyne laser linewidth measurements using Wiener filters
No full textSelf-heterodyne beat note measurements are widely used for the experimental characterization of the frequency noise power spectral density (FN-PSD) and the spectral linewidth of lasers. The measured data, however, must be corrected for the transfer function of the experimental setup in a post-processing routine. The standard approach disregards the detector noise and thereby induces reconstruction artifacts, i.e., spurious spikes, in the reconstructed FN-PSD. We introduce an improved post-processing routine based on a parametric Wiener filter that is free from reconstruction artifacts, provided a good estimate of the signal-to-noise ratio is supplied. Building on this potentially exact reconstruction, we develop a new method for intrinsic laser linewidth estimation that is aimed at deliberate suppression of unphysical reconstruction artifacts. Our method yields excellent results even in the presence of strong detector noise, where the intrinsic linewidth plateau is not even visible using the standard method. The approach is demonstrated for simulated time series from a stochastic laser model including 1/f-type noise
The throughput in multi-channel (slotted) ALOHA: Large deviations and analysis of bad events
Get PDFWe consider ALOHA and slotted ALOHA protocols as medium access rules for a multi-channel message delivery system. Users decide randomly and independently with a minimal amount of knowledge about the system at random times to make a message emission attempt. We consider the two cases that the system has a fixed number of independent available channels, and that interference constraints make the delivery of too many messages at a time impossible. We derive probabilistic formulas for the most important quantities like the number of successfully delivered messages and the number of emission attempts, and we derive large-deviation principles for these quantities in the limit of many participants and many emission attempts. We analyse the rate functions and their minimizers and derive laws of large numbers for the throughput. We optimize it over the probability parameter. Furthermore, we are interested in questions like ``if the number of successfully delivered messages is significantly lower than the expectation, was the reason that too many or too few sending attempts were made?''. Our main tools are basic tools from probability and the theory of (the probabilities of) large deviations
Temporal dissipative solitons in the Morris-Lecar model with time-delayed feedback
No full textWe study the dynamics and bifurcations of temporal dissipative solitons in an excitable system under time-delayed feedback. As a prototypical model displaying different types of excitability, we use the Morris-Lecar model. In the limit of large delay, soliton like solutions of delay-differential equations can be treated as homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of pulse solutions and to explain their dependence on the system parameters. In particular, we show how a homoclinic orbit flip of a single-pulse soliton leads to the destabilization of equidistant multi-pulse solutions and to the emergence of stable pulse packages. It turns out that this transition is induced by a heteroclinic orbit flip in the system without feedback, which is related to the excitability properties of the Morris-Lecar model
Approximating dynamic phase-field fracture in viscoelastic materials with a first-order formulation for velocity and stress
Get PDFWe investigate a model for dynamic fracture in viscoelastic materials at small strains. While the sharp crack interface is approximated with a phase-field method, we consider a viscous evolution with a quadratic dissipation potential for the phase-field variable. A non-smooth constraint enforces a unidirectional evolution of the phase-field, i.e. material cannot heal. The viscoelastic equation of motion is transformed into a first order formulation and coupled in a nonlinear way to the non-smooth evolution law of the phase field. The system is fully discretized in space and time with a discontinuous Galerkin approach for the first-order formulation. Based on this, existence of discrete solutions is shown and, as the step size in space and time tends to zero, their convergence to a suitable notion of weak solution of the system is discussed
Trotter-type formula for operator semigroups on product spaces
Get PDFWe consider a Trotter-type-product formula for approximating the solution of a linear abstract Cauchy problem (given by a strongly continuous semigroup), where the underlying Banach space is a product of two spaces. In contrast to the classical Trotter-product formula, the approximation is given by freezing subsequently the components of each subspace. After deriving necessary stability estimates for the approximation, which immediately provide convergence in the natural strong topology, we investigate convergence in the operator norm. The main result shows that an almost optimal convergence rate can be established if the dominant operator generates a holomorphic semigroup and the off-diagonal coupling operators are bounded