
Tuesday, June 16, 2009
Colloquium "50 Years Numerische Mathematik"
9:30 a.m.  6 p.m., Leibniz Supercomputing Centre, Lecture Hall, Boltzmannstraße 1
Preliminary schedule:

9:30 Alfio Quarteroni
École Polytechnique Fédérale de Lausanne
Politecnico di Milano
Numerical Modeling through Domain Decomposition and Applications
Mathematical models of multiphysics problems can be conveniently accommodated in the framework of domain splitting. In this presentation I will introduce a general mathematical setting, discuss how domain decomposition algorithms and preconditioners can be called into play, then I will address some applications to the field of cardiovascular modeling and that of design and simulation for sports competition.

10:00 Douglas N. Arnold
University of Minnesota at Minneapolis
50 Years of Whitney Forms
Like Numerische Mathematik itself, the story of the Whitney forms
(or Whitney elements) began about 50 years ago, has flourished
since, and continues to be very active. In 1957 Hassler Whitney
introduced these spaces of piecewise linear differential
forms in order to study problems on the interface of algebraic
topology and differential geometry. The Whitney forms have played
a role in geometry ever since, including leading to the solution
of an important conjecture. Independently, in 1975 Raviart and
Thomas introduced their famous mixed finite elements, which were
generalized to three dimensions in two separate ways by Nédélec
in 1980, as the edge elements and the face elements. In 1988
Bossavit pointed out that these finite elements are exactly the
Whitney forms. Whitney elements have proved to be hugely useful,
and are particularly widely adopted by the practitioners in the
computational electromagnetics community, who are now much greater
users of them than the geometers. In this talk we will
explain the origin and use of the Whitney forms in geometry, and
describe how an understanding of their geometrical underpinnings
is enabling major progress in numerical mathematics.

10:30 Marc A. Schweitzer
Universität Bonn
Meshfree Multilevel Methods for Partial Differential Equations
Meshfree methods have enjoyed a significant research effort in the past
20 years and substantial improvements have been
accomplished. Many different numerical schemes have been proposed. For
instance, the Diffuse Element Method, Smoothed
Particle Hydrodynamics, Generalized Finite Difference Method, Radial
Basis Functions, Reproducing Particle Kernel
Method, Element Free Galerkin Method, Meshless Local Petrov Galerkin
Method, Generalized/eXtended Finite Element
Methods, or Partition of Unity Methods.
In this talk we review the two key concepts employed in the construction
of the trial space of many meshfree Galerkin
methods, the partition of unity approach and enrichment, which allow for
the straightforward incorporation of
nonpolynomial shape functions in the approximation space. Thus, a
priori information about singular or discontinuous
behavior of the solution of a PDE can be encoded implicitly in the trial
space and must not resolved by (adaptive) mesh
refinement; i.e. we can employ an algebraic instead of geometric
refinement of the PUM trial spaces which leads to a
smaller number of unknowns. This is especially advantageous in a
multilevel setting since singularities and
discontinuities of the solution can be resolved easily on all levels. We
apply the multilevel particlepartition of
unity method to some reference problems from fracture mechanics to
demonstrate the overall efficiency of this algebraic
refinement approach.
 11:00 Coffee break

11:30 Wolfgang Dahmen
RWTH Aachen
Compressed Sensing  Near Optimal Recovery of Signals from
Highly Incomplete Measurements
The usual paradigm for signal processing is to model a signal as a bandlimited function and capture it signal by means of its time samples. The ShannonNyquist theory says that the sampling rate needs to be at least twice the bandwidth. For broadbanded signals, such high sampling rates may be impossible to implement in circuitry. Compressed Sensing is a new area of signal processing whose aim is to circumvent this dilemma by sampling signals closer to their information rate instead of their bandwidth. Rather than model the signal as bandlimited, Compressed Sensing assumes that the signal can be represented or approximated by a few suitably chosen terms from a basis expansion of the signal. It also enlarges the concept of sample to include the application of any linear functional applied to the signal.
We give a brief introduction to Compressed Sensing that centers on the effectiveness and implementation of random sampling.
After briefly touching on the mathematical background,
we discuss the notion of instance optimality
as a performance benchmark that applies also to nonsparse signals. We sketch
instance optimal decoding techniques with special emphasis on and thresholding techniques.

12:00 Carl W. R. de Boor
University of Wisconsin at Madison
Issues in Multivariate Polynomial Interpolation
 12:30 Lunch

14:10 G. W. (Pete) Stewart
University of Maryland
The Semidefinite BArnoldi Algorithm

14:40 Olof B. Widlund
New York University
Recent Advances on Domain Decomposition Algorithms for Almost Incompressible Elasticity
The domain decomposition methods considered are preconditioned
conjugate gradient methods designed for the very large algebraic
systems of equations which often arise in finite element practice.
They are designed for massively parallel computer systems and the
preconditioners are built from solvers on the substructures into which
the domain of the given problem is partitioned. In addition, to
obtain scalability, there must be a coarse problem, with a small
number of degrees of freedom for each substructure. The design
of this coarse problem is crucial for obtaining rapidly convergent
iterations and poses the most interesting challenge in the analysis.
Results for two families of domain decomposition methods from the
overlapping Schwarz and the FETIDP/BDDC families will be discussed
with a special emphasis on almost incompressible elasticity approximated
by mixed finite element methods. Some of these algorithms are now
used extensively at the SANDIA, Albuquerque laboratories and will soon
be made available as public domain software.
This work is being carried out in close collaboration with
Clark R. Dohrmann of the Sandia National Laboratories,
Albuquerque, NM and Axel Klawonn and Oliver Rheinbach
of the University of DuisburgEssen, Germany.
 15:10 Coffee break

15:40 Endre Süli
University of Oxford
Mathematical Challenges in Kinetic Models of Dilute Polymers
The purpose of this talk is to review recent analytical and computational results for macroscopicmicroscopic beadspring models that arise from the kinetic theory of dilute solutions of incompressible polymeric fluids with noninteracting polymer chains, involving the coupling of the unsteady NavierStokes system in a bounded domain Ω ⊂R^{d}, d=2 or 3, with an elastic extrastress tensor as righthand side in the momentum equation, and a (possibly degenerate) FokkerPlanck equation over the (2d+1)dimensional region &Omega x D x [0,T], where D ⊂ R^{d} is the configuration domain and [0,T] is the temporal domain. The FokkerPlanck equation arises from a system of (Itô) stochastic differential equations, which models the evolution of a 2dcomponent vectorial stochastic process comprised by the dcomponent centreofmass vector and the dcomponent orientation (or configuration) vector of the polymer chain. We show the existence of globalintime weak solutions to the coupled NavierStokesFokkerPlanck system for a general class of spring potentials including, in particular, the widely used finitely extensible nonlinear elastic (FENE) potential. The numerical approximation of this highdimensional coupled system is a formidable computational challenge, complicated by the fact that for practically relevant spring potentials, such as the FENE potential, the drift term in the FokkerPlanck equation is unbounded on ∂D.

16:10 Michael J. Holst
University of California at San Diego
Local Convergence of Adaptive Methods for Nonlinear Equations
In this talk we develop a convergence framework for an abstract adaptive
finite elementlike algorithm for nonlinear operator equations in lpBanach
presheaves over an underlying measure space (essentially Banach spaces with
local structure). We first develop the convergence framework for nonlinear
operators which are locally Lipschitz and satisfy a local infsup condition,
giving a general convergence result. We next introduce some additional
conditions that allow for an improvement of the convergence framework to one
that ensures contraction. We then indicate how the convergence framework can
be used to recover a number of existing convergence and optimality results
for linear and nonlinear elliptic problems on open sets in R^{n}, as well as
establish some new results for geometric elliptic PDE problems posed on
Riemannian manifolds. The abstract convergence framework we develop helps
clarify some of the core common ideas present in convergence analysis for
adaptive methods.
17:00 Tour to the supercomputing facilities at LRZ

