Freie Universität Berlin
Repository: Freie Universität Berlin (FU), Math Department (fu_mi_publications)Not a member yet
2251 research outputs found
Sort by
Sharp-interface problem of the Ohta-Kawasaki model for symmetric diblock copolymers
No full textThe Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space
The climatology and nature of warm-season convective cells in cold-frontal environments over Germany
No full textCold fronts provide an environment particularly favourable for convective initiation in the mid-latitudes and can also be associated with convective hazards such as flooding, wind, hail and lightning. We build a climatology of cold-frontal convective cells between 2007–2016 for April–September in a cell-front distance framework by combining a radar-based cell detection and tracking dataset and automatic front detection methods applied to reanalysis data. We find that on average around twice as many cells develop on cold-frontal cell days compared to non-cold-frontal cell days. Using the 700 hPa level as a reference point, we show the maximum cell frequency is 350–400 km ahead of the 700 hPa front, which is marginally ahead of the typical surface front location. The 700 hPa front location marks the minimum cell frequency and a clear shift in regime between cells, with a weakened diurnal cycle on the warm side of the 700 hPa cold front and strongly diurnally driven cells on the cold side of the 700 hPa front. High cell frequency is found several hundreds of kilometres ahead of the surface front, and cells in this region are most likely to be associated with mesocyclones, intense convective cores and lightning. Namely, mesocyclones were detected in around 5.0 % of pre-surface-frontal cells compared to only 1.5 % of non-cold-frontal cells. The findings in this study are an important step towards a better understanding of cold-frontal convection climatology and links between cold fronts and convective hazards
Den Ballast tradierter Zerrbilder in Mathematikschulbüchern endlich abwerfen
No full textDiskurse über Gestaltung von Mathematikschulbüchern widmen sich bislang vor allem der Frage,inwieweit die in Aufgabenstellungen fachdidaktisch aufbereiteten, fachwissenschaftlichen Themen undInhalte einem modernen, kompetenzorientierten Mathematikunterricht entsprechen. Teilweise wird indiesem Zusammenhang auch thematisiert, ob durch Aufgabenformate oder Aufgabenkontexte einadäquates Bild der Mathematik gezeigt und vermittelt wird. Beide Aspekte sind unbestritten wichtig undnotwendig. Zu wenig Beachtung finden jedoch bislang andere Perspektiven, die sich ebenfalls kritisch miteiner zeitgemäßen Gestaltung von Mathematikschulbüchern auseinandersetzen und eben auch Bestandteilinnerfachlicher Diskussionen sind. Zu ihnen gehört die genderorientierte Schulbuchforschun
Predicting Coma Recovery After Cardiac Arrest With Residual Neural Networks
No full textAims
Interpretation of continuous EEG is a demanding task that requires the expertise of trained neurologists. However, these experts are not always available in many medical centers. As part of the 2023 George B. Moody PhysioNet Challenge, we developed a deep learning based method for analyzing EEG data of comatose patients and predicting prognosis following cardiac arrest.
Methods
Our approach is a two-step pipeline that consists of a prediction model and a decision-making strategy. The prediction model is a residual neural network (ResNet-18) that extracts features and makes a prediction based on a short 5-minute EEG recording. In the second step, a majority vote over multiple predictions made for several EEG recordings of a patient determines the final prognosis
Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise
No full textWe prove the positivity of Lyapunov exponents for the normal form of a Hopf bifurcation, perturbed by additive white noise, under sufficiently strong shear strength. This completes a series of related results for simplified situations which we can exploit by studying suitable limits of the shear and noise parameters. The crucial technical ingredient for making this approach rigorous is a result on the continuity of Lyapunov exponents via Furstenberg–Khasminskii formulas
Overcoming the Timescale Barrier in Molecular Dynamics: Transfer Operators, Variational Principles, and Machine Learning
No full textOne of the main challenges in molecular dynamics is overcoming the ‘timescale barrier’: in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behaviour on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory, as well as the algorithmic development, from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in molecular dynamics. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject
Detecting the birth and death of finite-time coherent sets
Get PDFFinite-time coherent sets (FTCSs) are distinguished regions of phase space that resist mixing with the surrounding space for some finite period of time; physical manifestations include eddies and vortices in the ocean and atmosphere, respectively. The boundaries of FTCSs are examples of Lagrangian coherent structures (LCSs). The selection of the time duration over which FTCS and LCS computations are made in practice is crucial to their success. If this time is longer than the lifetime of coherence of individual objects then existing methods will fail to detect the shorter-lived coherence. It is of clear practical interest to determine the full lifetime of coherent objects, but in complicated practical situations, for example a field of ocean eddies with varying lifetimes, this is impossible with existing approaches. Moreover, determining the timing of emergence and destruction of coherent sets is of significant scientific interest. In this work we introduce new constructions to address these issues. The key components are an inflated dynamic Laplace operator and the concept of semi-material FTCSs. We make strong mathematical connections between the inflated dynamic Laplacian and the standard dynamic Laplacian, showing that the latter arises as a limit of the former. The spectrum and eigenfunctions of the inflated dynamic Laplacian directly provide information on the number, lifetimes, and evolution of coherent sets
A 'periodic table' of modes and maximum a posteriori estimators
No full textThe last decade has seen many attempts to generalise the definition of modes, or MAP estimators, of a probability distribution μ on a space X to the case that μ has no continuous Lebesgue density, and in particular to infinite-dimensional Banach and Hilbert spaces X. This paper examines the properties of and connections among these definitions. We construct a systematic taxonomy -- or `periodic table' -- of modes that includes the established notions as well as large hitherto-unexplored classes. We establish implications between these definitions and provide counterexamples to distinguish them. We also distinguish those definitions that are merely `grammatically correct' from those that are `meaningful' in the sense of satisfying certain `common-sense' axioms for a mode, among them the correct handling of discrete measures and those with continuous Lebesgue densities. However, despite there being 17 such `meaningful' definitions of mode, we show that none of them satisfy the `merging property', under which the modes of μ|A, μ|B and μ|A∪B enjoy a straightforward relationship for well-separated positive-mass events A,B⊆X
Rough McKean–Vlasov dynamics for robust ensemble Kalman filtering
No full textMotivated by the challenge of incorporating data into misspecified and multiscale dynamical models, we study a McKean–Vlasov equation that contains the data stream as a common driving rough path. This setting allows us to prove well-posedness as well as continuity with respect to the driver in an appropriate rough-path topology. The latter property is key in our subsequent development of a robust data assimilation methodology: We establish propagation of chaos for the associated interacting particle system, which in turn is suggestive of a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework. Finally, we discuss a data-driven method based on subsampling to construct suitable rough path lifts and demonstrate the robustness of our scheme in a number of numerical experiments related to parameter estimation problems in multiscale contexts
Reduced Basis Approach for Convection-Diffusion Equations with Non-linear Boundary Reaction Conditions
No full textThis paper aims at an efficient strategy to solve drift-diffusion problems with non-linear boundary conditions as they appear, e.g., in heterogeneous catalysis. Since the non-linearity only involves the degrees of freedom along (a part of) the boundary, a reduced basis ansatz is suggested that computes discrete Green’s-like functions for the present drift-diffusion operator such that the global non-linear problem reduces to a smaller non-linear problem for a boundary method. The computed basis functions are completely independent of the non-linearities. Thus, they can be reused for problems with the same differential operator and geometry. Corresponding scenarios might be inverse problems in heterogeneous catalysis but also modeling the effect of different catalysts in the same reaction chamber. The strategy is explained for a mass-conservative finite volume method and demonstrated on a simple numerical example for catalytic CO oxidation