Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Frank Filbir
Institut für Biomathematik und Biometrie, Helmholtz Zentrum München
Approximation on manifolds
In many practical applications, for example document analysis, semi-supervised learning, and inverse
problems one is confronted with functions defined on a (Riemannian) manifold M imbedded in a high
dimensional ambient space. These functions have to be approximated by using sample values of the
function. Due to several restrictions like experimental setup etc. we can hardly assume that the sampling
nodes are located on a regular grid. This means we have to come up with an approximation process which
can, on the one hand, work with scattered data and, on the other hand, has sufficiently good approximation
rate. We consider approximation processes of the form
σL f (x) =
∞
∑
j=0
H �ℓj �
〈 f ,φj〉φ j(x),
L
where H is a suitable filter function and {φ j} is an orthonormal function system on the manifold M. In order
to get an approximation process which has the aforementioned properties it is necessary to construct
quadrature formulas with certain degree of exactness. In this talk we will address this problem and we will
show how this is related to the problem of constructing well localized kernels on M. This talk is based on
joint work with Hrushikesh N. Mhaskar, Department of Mathematics, California State University, U.S.A.
Michael Pippig
Technische Universität Chemnitz
Parallel fast fourier transforms and their application to particle simulation
The direct computation of Coulomb interactions in large particle systems is a computational demanding
problem. For periodic boundary conditions, Ewald proposed to split the interactions into two fast converging
parts. While the first part is short ranged and includes the singularity, the long ranged and smooth
part converts fast in Fourier domain. Using nonequispaced fast Fourier transforms, the calculation of the
smooth part can be further sped up. This leads to a fast algorithm comparable to the particle-particle
particle-mesh method. During this talk, we develop an algorithm for the massively parallel computation of
the equispaced fast Fourier transform, generalize it to the nonequispaced case and apply it to parallelize
the fast Ewald summation. The resulting algorithm will be compared to other algorithms for the fast
computation of Coulomb interactions.
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