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Classification and clustering: models, software and applications
We are pleased to present the report on the 30th Fall Meeting of the working group ``Data Analysis and Numerical Classification'' (AG-DANK) of the German Classification Society. The meeting took place at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, from Friday Nov. 14 till Saturday Nov. 15, 2008. Already 12 years ago, WIAS had hosted a traditional Fall Meeting with special focus on classification and multivariate graphics (Mucha and Bock, 1996). This time, the special topics were stability of clustering and classification, mixture decomposition, visualization, and statistical software.
Impact of slippage on the morphology and stability of a dewetting rim
In this study lubrication theory is used to describe the stability and morphology of the rim that forms as a thin polymer film dewets from a hydrophobized silicon wafer. Thin film equations are derived from the governing hydrodynamic equations for the polymer to enable the systematic mathematical and numerical analysis of the properties of the solutions for different regimes of slippage and for a range of time scales. Dewetting rates and the cross sectional profiles of the evolving rims are derived for these models and compared to experimental results. Experiments also show that the rim is typically unstable in the spanwise direction and develops thicker and thinner parts that may grow into ``fingers''. Linear stability analysis as well as nonlinear numerical solutions are presented to investigate shape and growth rate of the rim instability. It is demonstrated that the difference in morphology and the rate at which the instability develops can be directly attributed to the magnitude of slippage. Finally, a derivation is given for the dominant wavelength of the bulges along the unstable rim.
A Redlich-Kister type free energy model for Li-intercalation compounds with variable lattice occupation numbers
One of the central quantities of a lithium ion intercalation compound is the open circuit potential, the voltage a battery material delivers in thermodynamic equilibrium. This voltage is related to the chemical potential of lithium in the insertion material and in general a non-linear function of the mole fraction of intercalated lithium. Experimental data shows further that it is specific for various materials. The open circuit voltage is a central ingredient for mathematical models of whole battery cells, which are used to investigate and simulate the charge and discharge behavior and to interpret experimental data on non-equilibrium processes. However, since no overall predictive theoretical method presently exists for the open circuit voltage, it is commonly fitted to experimental data. Simple polynomial fitting approaches are widely used, but they lack any thermodynamic interpretation. More recently systematically and thermodynamically motivated approaches are used to model the open circuit potential. We provide here an explicit free energy density which accounts for variable occupation numbers of Li on the intercalation lattice as well as RedlichKister-type enthalpy contributions. The derived chemical potential is validated by experimental data of Liy(Ni1/3Mn1/3Co1/3)O2 and we show that only two parameters are sufficient to obtain an overall agreement of the non-linear open circuit potential within the experimental error.
On the evolution of the empirical measure for the hard-sphere dynamics
We prove that the evolution of marginals associated to the empirical measure of a finite system of hard spheres is driven by the BBGKY hierarchical expansion. The usual hierarchy of equations for L1 measures is obtained as a corollary. We discuss the ambiguities arising in the corresponding notion of microscopic series solution to the Boltzmann-Enskog equation.
Optimal and robust a posteriori error estimates in L∞(L2) for the approximation of Allen-Cahn equations past singularities
Optimal a posteriori error estimates in L∞(L2) are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.
Anisotropic finite element mesh adaptation via higher dimensional embedding
In this paper we provide a novel anisotropic mesh adaptation technique for adaptive finite element analysis. It is based on the concept of higher dimensional embedding, which was exploited in [1-4] to obtain an anisotropic curvature adapted mesh that fits a complex surface in ℝ3. In the context of adaptive finite element simulation, the solution (which is an unknown function ƒ: Ω ⊂; ℝd → ℝ) is sought by iteratively modifying a finite element mesh according to a mesh sizing field described via a (discrete) metric tensor field that is typically obtained through an error estimator. We proposed to use a higher dimensional embedding, Φf(x) := (x1, …, xd, s f (x1, …, xd), s ∇ f (x1, …, xd))t, instead of the mesh sizing field for the mesh adaption. This embedding contains both informations of the function ƒ itself and its gradient. An isotropic mesh in this embedded space will correspond to an anisotropic mesh in the actual space, where the mesh elements are stretched and aligned according to the features of the function ƒ. To better capture the anisotropy and gradation of the mesh, it is necessary to balance the contribution of the components in this embedding. We have properly adjusted Φf(x) for adaptive finite element analysis. To better understand and validate the proposed mesh adaptation strategy, we first provide a series of experimental tests for piecewise linear interpolation of known functions. We then applied this approach in an adaptive finite element solution of partial differential equations. Both tests are performed on two-dimensional domains in which adaptive triangular meshes are generated. We compared these results with the ones obtained by the software BAMG - a metric-based adaptive mesh generator. The errors measured in the L2 norm are comparable. Moreover, our meshes captured the anisotropy more accurately than the meshes of BAMG.
Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential
A fully computable upper bound for the finite element approximation error of Allen-Cahn and Cahn-Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.
A Bayesian approach to parameter identification in gas networks
The inverse problem of identifying the friction coefficient in an isothermal semilinear Euler system is considered. Adopting a Bayesian approach, the goal is to identify the distribution of the quantity of interest based on a finite number of noisy measurements of the pressure at the boundaries of the domain. First well-posedness of the underlying non-linear PDE system is shown using semigroup theory, and then Lipschitz continuity of the solution operator with respect to the friction coefficient is established. Based on the Lipschitz property, well-posedness of the resulting Bayesian inverse problem for the identification of the friction coefficient is inferred. Numerical tests for scalar and distributed parameters are performed to validate the theoretical results.
Auditory cortex modelled as a dynamical network of oscillators: Understanding event-related fields and their adaptation
Adaptation, the reduction of neuronal responses by repetitive stimulation, is a ubiquitous feature of auditory cortex (AC). It is not clear what causes adaptation, but short-term synaptic depression (STSD) is a potential candidate for the underlying mechanism. We examined this hypothesis via a computational model based on AC anatomy, which includes serially connected core, belt, and parabelt areas. The model replicates the event-related field (ERF) of the magnetoencephalogram as well as ERF adaptation. The model dynamics are described by excitatory and inhibitory state variables of cell populations, with the excitatory connections modulated by STSD. We analysed the system dynamics by linearizing the firing rates and solving the STSD equation using time-scale separation. This allows for characterization of AC dynamics as a superposition of damped harmonic oscillators, so-called normal modes. We show that repetition suppression of the N1m is due to a mixture of causes, with stimulus repetition modifying both the amplitudes and the frequencies of the normal modes. In this view, adaptation results from a complete reorganization of AC dynamics rather than a reduction of activity in discrete sources. Further, both the network structure and the balance between excitation and inhibition contribute significantly to the rate with which AC recovers from adaptation. This lifetime of adaptation is longer in the belt and parabelt than in the core area, despite the time constants of STSD being spatially constant. Finally, we critically evaluate the use of a single exponential function to describe recovery from adaptation.
Anisotropic growth of random surfaces in 2 + 1 dimensions
We construct a family of stochastic growth models in $2+1$ dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield $1+1$ dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models. The main asymptotic results are: (1) The growing surface has a limit shape that consists of facets interpolated by a curved piece. (2) The one-point fluctuations of the height function in the curved part are asymptotically normal with variance of order $ln(t)$ for time $tgg 1$. (3) There is a map of the $(2+1)$-dimensional space-time to the upper half-plane $H$ such that on space-like submanifolds the multi-point fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on $H$.