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Geometric multiscale decompositions of dynamic low-rank matrices
(2012)SAM Research ReportThe present paper is concerned with the study of manifold-valued multiscale transforms with a focus on the Stiefel manifold. For this specific geometry we derive several formulas and algorithms for the computation of geometric means which will later enable us to construct multiscale transforms of wavelet type. As an application we study compression of piecewise smooth families of low-rank matrices both for synthetic data and also real-world ...Report -
Polar Spectral Scheme for the Spatially Homogeneous Boltzmann Equation
(2014)SAM Research ReportWe consider the non-linear spatially homogeneous Boltzmann equation, and develop a polar spectral discretization in two dimensions based on Laguerre polynomials, which generalizes previous methods by Ender and Ender [A.Ya.~Ender and I.A.~Ender: Polynomial expansions for the isotropic Boltzmann equation and invariance of the collision integral with respect to the choice of basis functions. Physics of Fluids, 11:2720--2730, 1999] to the ...Report -
Finite elements of arbitrary order and quasiinterpolation for data in Riemannian manifolds
(2011)SAM Research ReportWe consider quasiinterpolation operators for functions assuming their values in a Riemannian manifold. We construct such operators from corresponding linear quasiinterpolation operators by replacing affine averages with the Riemannian center of mass. As a main result we show that the approximation rate of such a nonlinear operator is the same as for the linear operator it has been derived from. In order to formulate this result in an ...Report -
Parabolic Molecules: Curvelets, Sherlets and Beyond
(2013)SAM Research ReportAnisotropic representation systems such as curvelets and shearlets have had a significant impact on applied mathematics in the last decade. The main reason for their success is their superior ability to optimally resolve anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shock fronts in solutions of transport dominated equations. By now, a large variety of ...Report -
Tensor-product discretization for the spatially inhomogeneous and transient Boltzmann equation in 2D
(2015)SAM Research ReportIn this paper we extend the previous work [E. Fonn, P. Grohs, and R. Hiptmair, Polar spectral scheme for the spatially homogeneous Boltzmann equation, Tech. Rep. 2014-13, Seminar for Applied Mathematics, ETH Zurich, 2014.] for the homogeneous nonlinear Boltzmann equation to the spatially inhomogeneous case. We employ a (Petrov)-Galerkin discretization in the velocity variable of the Boltzmann collision operator based on Laguerre polynomials ...Report -
Continuous Parabolic Molecules
(2015)Research ReportDecomposition systems based on parabolic scaling have in the last years garnered attention for their ability to answer questions regarding curvilinear singularities of functions. Well known examples of these systems are curvelets and shearlets. In recent years there has been a sufficient body of evidence to suggest that these systems are able to answer the same fundamental questions and it should thus be possible to consider them as parts ...Report -
Discrete deep feature extraction: A theory and new architectures
(2016)SAM Research ReportReport -
$\alpha$-Molecules
(2013)Research ReportThe novel framework of parabolic molecules provides for the first time a unifying framework for (sparse) approximation properties of directional representation systems by, in particular, including curvelets and shearlets. However, the considered common bracket is parabolic scaling, which excludes systems such as ridgelets and wavelets. In this paper, we therefore provide a generalization of this framework, which we coin α-molecules, by ...Report