The colloquium of the Modeling, Numerics, Differential Equations Group at the institute of mathematics is a conventional colloquium attracting a broad audience consisting of professors and research assistants of all associated work groups, in particular applied functional analysis, numerical linear algebra, and partial differential equations. Graduate students are also attending the colloquium.

For these reasons we look forward to talks aimed at non-specialists that can be be enjoyed by graduate students.

Ronald W. Hoppe (Universität Augsburg / University of Houston)

Numerical Solution of Second and Fourth Order Total Variation Flow Problems

María Soledad Aronna (Escola de Matemática Aplicada Rio de Janeiro)

In this talk I will present the main features of first and second order optimality conditions for optimal control problems of ordinary differential equations that are affine with respect to the control and nonlinear with respect to the state. Assuming the presence of control constraints, I will present a second order sufficient condition for optimality and a numerical scheme in the form of a shooting method.

David Siska (University of Edinburgh)

Exponential Convergence of Policy Improvement Algorithm for Controlled Diffusions

In this talk I will present the main features of first and second order optimality conditions for optimal control problems of ordinary differential equations that are affine with respect to the control and nonlinear with respect to the state. Assuming the presence of control constraints, I will present a second order sufficient condition for optimality and a numerical scheme in the form of a shooting method.

Herbert Egger (TU Darmstadt)

On the systematic approximation of evolution problems
with dissipation, Hamiltonian, or gradient structure.

A general framework for the numerical approximation of evolution
problems is presented that allows
to preserve an underlying dissipative, Hamiltonian, or gradient
structure. The approach is based on rewriting
the evolution problem in a particular form that complies with the
underlying structure and its variational
formulation. The underlying structure is then preserved automatically
under Galerkin projection in space,
which allows to deduce important structural properties for appropriate
discretization schemes including
projection based model reduction methods.

For the time-discretization, we consider two different approaches
depending on the underlying geometric
structure, i.e., discontinuous Galerkin and Petrov-Galerkin
approximations. Again, the basic structure of
the problem is inherited automatically by the proposed discretization
schemes.

The presented framework is rather general and allows the numerical
approximation of a wide range of applications,
including nonlinear partial differential equations and port-Hamiltonian
systems. Several examples will be discussed
for illustration and some connections to other discretization approaches
will be revealed.