Institute For Numerische Und Angewandte Mathematik

Institute For Numerische Und Angewandte Mathematik

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Fournier D.,Institute For Numerische Und Angewandte Mathematik | Gizon L.,Max Planck Institute for Solar System Research | Gizon L.,University of Gottingen | Hohage T.,Institute For Numerische Und Angewandte Mathematik | Birch A.C.,Max Planck Institute for Solar System Research
Astronomy and Astrophysics | Year: 2014

Context. In time-distance helioseismology, information about the solar interior is encoded in measurements of travel times between pairs of points on the solar surface. Travel times are deduced from the cross-covariance of the random wave field. Here, we consider travel times and also products of travel times as observables. They contain information about the statistical properties of convection in the Sun. Aims. We derive analytic formulae for the noise covariance matrix of travel times and products of travel times. Methods. The basic assumption of the model is that noise is the result of the stochastic excitation of solar waves, a random process that is stationary and Gaussian. We generalize the existing noise model by dropping the assumption of horizontal spatial homogeneity. Using a recurrence relation, we calculate the noise covariance matrices for the moments of order 4, 6, and 8 of the observed wave field, for the moments of order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and 4 of the travel times. Results. All noise covariance matrices depend only on the expectation value of the cross-covariance of the observed wave field. For products of travel times, the noise covariance matrix consists of three terms proportional to 1 /T, 1 /T2, and 1 /T 3, where T is the duration of the observations. For typical observation times of a few hours, the term proportional to 1 /T2 dominates and Cov [τ1τ2, τ3τ4] ≈ Cov [τ1, τ3] Cov [τ2,τ4] + Cov [τ1,τ4] Cov [τ2,τ3], where the τi are arbitrary travel times. This result is confirmed for p1 travel times by Monte Carlo simulations and comparisons with SDO/HMI observations. Conclusions. General and accurate formulae have been derived to model the noise covariance matrix of helioseismic travel times and products of travel times. These results could easily be generalized to other methods of local helioseismology, such as helioseismic holography and ring diagram analysis. © 2014 ESO.

Wolfram M.-T.,Wilberforce Road | Burger M.,Institute For Numerische Und Angewandte Mathematik | Siwy Z.S.,University of California at Irvine
Journal of Physics Condensed Matter | Year: 2010

High surface charges of polymer pore walls and applied electric fields can lead to the formation and subsequent dissolution of precipitates in nanopores. These precipitates block the pore, leading to current fluctuations. We present an extended Poisson-Nernst-Planck system which includes chemical reactions of precipitation and dissolution. We discuss the mathematical modeling and present 2D numerical simulations. © 2010 IOP Publishing Ltd.

Mues T.,Institute For Physikalische Chemie | Heuer A.,Institute For Physikalische Chemie | Burger M.,CeNoS | Burger M.,Institute For Numerische Und Angewandte Mathematik | And 2 more authors.
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics | Year: 2010

Oscillatory zoning (OZ) occurs in all major classes of minerals and also in a wide range of geological environments. It is caused by self-organization and describes fluctuations of the spatial chemical composition profile of the crystal. We present here a two-dimensional model of OZ based on our previous one-dimensional (1D) analysis and investigate whether the results of the 1D stability analysis remain valid. With the additional second dimension we were able to study the origin of the spatially homogeneous layer formation by linear stability analysis. Numerical solutions of the model are presented and the results of a Fourier analysis delivers a detailed understanding of the crystal growth behavior as well as the limits of the model. Effects beyond linear stability analysis are important to finally understand the final structure formation. © 2010 The American Physical Society.

Bauer U.,Institute For Numerische Und Angewandte Mathematik | Schonlieb C.-B.,Institute For Numerische Und Angewandte Mathematik | Wardetzky M.,Institute For Numerische Und Angewandte Mathematik
AIP Conference Proceedings | Year: 2010

We present first insights into the relation between two popular yet apparently dissimilar approaches to denoising of one dimensional signals, based on (i) total variation (TV) minimization and (ii) ideas from topological persistence. While a close relation between (i) and (ii) might phenomenologically not be unexpected, our work appears to be the first to make this connection precise for one dimensional signals. We provide a link between (i) and (ii) that builds on the equivalence between TV-L 2 regularization and taut strings and leads to a novel and efficient denoising algorithm that is contrast preserving and operates in O(nlogn) time, where n is the size of the input. © 2010 American Institute of Physics.

Engwer C.,Institute For Numerische Und Angewandte Mathematik | Hillen T.,University of Alberta | Knappitsch M.,Institute For Numerische Und Angewandte Mathematik | Surulescu C.,University of Kaiserslautern
Journal of Mathematical Biology | Year: 2015

Gliomas are a class of rarely curable tumors arising from abnormal glia cells in the human brain. The understanding of glioma spreading patterns is essential for both radiological therapy as well as surgical treatment. Diffusion tensor imaging (DTI) allows to infer the white matter fibre structure of the brain in a noninvasive way. Painter and Hillen (J Theor Biol 323:25–39, 2013) used a kinetic partial differential equation to include DTI data into a class of anisotropic diffusion models for glioma spread. Here we extend this model to explicitly include adhesion mechanisms between glioma cells and the extracellular matrix components which are associated to white matter tracts. The mathematical modelling follows the multiscale approach proposed by Kelkel and Surulescu (Math Models Methods Appl Sci 23(3), 2012). We use scaling arguments to deduce a macroscopic advection-diffusion model for this process. The tumor diffusion tensor and the tumor drift velocity depend on both, the directions of the white matter tracts as well as the binding dynamics of the adhesion molecules. The advanced computational platform DUNE enables us to accurately solve our macroscopic model. It turns out that the inclusion of cell binding dynamics on the microlevel is an important factor to explain finger-like spread of glioma. © 2014, Springer-Verlag Berlin Heidelberg.