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Pizarro L.,Saarland University | Pizarro L.,Imperial College London | Mrazek P.,UPEK Inc | Didas S.,Fraunhofer Institute For Techno Und Wirtschaftsmathematik | And 2 more authors.
International Journal of Computer Vision | Year: 2010

We propose a discrete variational approach for image smoothing consisting of nonlocal data and smoothness constraints that penalise general dissimilarity measures defined on image patches. One of such dissimilarity measures is the weighted L 2 distance between patches. In such a case we derive an iterative neighbourhood filter that induces a new similarity measure in the photometric domain. It can be regarded as an extended patch similarity measure that evaluates not only the patch similarity of two chosen pixels, but also the similarity of their corresponding neighbours. This leads to a more robust smoothing process since the pixels selected for averaging are more coherent with the local image structure. By slightly modifying the way the similarities are computed we obtain two related filters: The NL-means filter of Buades et al. (SIAM Multiscale Model. Simul. 4(2):490-530, 2005b) and the NDS filter of Mrázek et al. (Geometric Properties for Incomplete Data, Computational Imaging and Vision, vol. 31, pp. 335-352, Springer, Dordrecht, 2006). In fact, the proposed approach can be considered as a generalisation of the latter filter to the space of patches. We also provide novel insights into relations of the NDS filter with diffusion/regularisation methods as well as with some recently proposed graph regularisation techniques. We evaluate our method for the task of denoising greyscale and colour images degraded with Gaussian and salt-and-pepper noise, demonstrating that it compares very well to other more sophisticated approaches. © 2010 Springer Science+Business Media, LLC.

Schafer M.,Fraunhofer Institute For Techno Und Wirtschaftsmathematik | Frank M.,RWTH Aachen | Levermore C.D.,University of Maryland University College
Multiscale Modeling and Simulation | Year: 2011

In this paper, we investigate moment methods from a general point of view using an operator notation. This theoretical approach lets us explore the moment closure problem in more detail. This gives rise to a new idea, proposed in Levermore [Transition Regime Models for Radiative Transport, Presentation at the Institute for Pure & Applied Mathematics: Grand Challenge Problems in Computational Astrophysics, Workshop on Transfer Phenomena, 2005, http://www.ipam.ucla.edu/publications/pcaws4/pcaws4-5724.pdf, and Moment Closures for Radiative Transport, manuscript, 2009], of how to improve the well-known P N approximations. We systematically develop a diffusive correction to the P N equations from the operator formulation-the so-called D N approximation. We validate the new approach with numerical examples in one and two dimensions. © 2011 Society for Industrial and Applied Mathematics.

Iliev O.,Fraunhofer Institute For Techno Und Wirtschaftsmathematik | Iliev O.,Bulgarian Academy of Science | Lazarov R.,Bulgarian Academy of Science | Lazarov R.,Texas A&M University | Willems J.,Texas A&M University
Multiscale Modeling and Simulation | Year: 2011

We present a two-scale finite element method (FEM) for solving Brinkman's and Darcy's equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes' equations by Wang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy's equations. In order to reduce the "resonance error" and to ensure convergence to the global fine solution, the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems. © 2011 Society for Industrial and Applied Mathematics.

Klauer A.,Fraunhofer Institute For Techno Und Wirtschaftsmathematik
Journal of Applied Analysis | Year: 2014

The complex Bloch varieties and the associated Fermi curves of two-dimensional periodic Schrödinger operators with quasi-periodic boundary conditions are defined as complex analytic varieties, the Schrödinger potentials being from the Lorentz-Fourier space ?l∞,1. Then, an asymptotic analysis of the Fermi curves is performed. The decomposition of a Fermi curve into a compact part, an asymptotically free part, and thin handles, is recovered as expected. Furthermore, it is shown that the set of potentials whose associated Fermi curve has finite geometric genus is a dense subset of ?l∞,1. Moreover, the Fourier transforms of the potentials are locally isomorphic to perturbed Fourier transforms induced by the handles. Finally, an asymptotic family of parameters describing the sizes of the handles is introduced. These parameters are good candidates for describing parts of the space of all Fermi curves. © 2014 by Walter de Gruyter Berlin/Bosto.

Herty M.,University of Kaiserslautern | Mohringb J.,Fraunhofer Institute For Techno Und Wirtschaftsmathematik | Sachersa V.,University of Kaiserslautern
Mathematical Methods in the Applied Sciences | Year: 2010

We introduce a new model for gas dynamics in pipe networks by asymptotic analysis. The model is derived from the isothermal Euler equations. We present the derivation of the model as well as numerical results illustrating the validity and its properties. We compare the new model with existing models from the mathematical and engineering literature. We further give numerical results on a sample network. Copyright © 2009 John Wiley & Sons, Ltd.

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