77 research outputs found
Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems
The central objective of this paper is to develop reduced basis methods for
parameter dependent transport dominated problems that are rigorously proven to
exhibit rate-optimal performance when compared with the Kolmogorov -widths
of the solution sets. The central ingredient is the construction of
computationally feasible "tight" surrogates which in turn are based on deriving
a suitable well-conditioned variational formulation for the parameter dependent
problem. The theoretical results are illustrated by numerical experiments for
convection-diffusion and pure transport equations. In particular, the latter
example sheds some light on the smoothness of the dependence of the solutions
on the parameters
Torsion free groups with indecomposable holonomy group I
We study the torsion free generalized crystallographic groups with the
indecomposable holonomy group which is isomorphic to either a cyclic group of
order or a direct product of two cyclic groups of order .Comment: 22 pages, AMS-Te
Symmetries and reversing symmetries of toral automorphisms
Toral automorphisms, represented by unimodular integer matrices, are
investigated with respect to their symmetries and reversing symmetries. We
characterize the symmetry groups of GL(n,Z) matrices with simple spectrum
through their connection with unit groups in orders of algebraic number fields.
For the question of reversibility, we derive necessary conditions in terms of
the characteristic polynomial and the polynomial invariants. We also briefly
discuss extensions to (reversing) symmetries within affine transformations, to
PGL(n,Z) matrices, and to the more general setting of integer matrices beyond
the unimodular ones.Comment: 34 page
Thomas Decomposition and Nonlinear Control Systems
This paper applies the Thomas decomposition technique to nonlinear control
systems, in particular to the study of the dependence of the system behavior on
parameters. Thomas' algorithm is a symbolic method which splits a given system
of nonlinear partial differential equations into a finite family of so-called
simple systems which are formally integrable and define a partition of the
solution set of the original differential system. Different simple systems of a
Thomas decomposition describe different structural behavior of the control
system in general. The paper gives an introduction to the Thomas decomposition
method and shows how notions such as invertibility, observability and flat
outputs can be studied. A Maple implementation of Thomas' algorithm is used to
illustrate the techniques on explicit examples
On Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type
The theory of Poisson vertex algebras (PVAs) (Barakat et al. in Jpn J Math 4(2):141–252, 2009) is a good framework to treat Hamiltonian partial differential equations. A PVA consists of a pair of a differential algebra and a bilinear operation called th
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