We consider the analysis, development and implementation of reliable and
competitive algorithms
for the numerical integration of differential equations arising in
mechanics and control. In particular multi-body systems, rigid body
simulation (of interest in satellite dynamics and celestial
mechanics), highly oscillatory systems (relevant for example in
modeling vibration of risers
descending form an oil platform).
Many of these problems share a common feature:
they possess an underlying geometric structure that has significant importance on the dynamical behavior
of the system, as it generally corresponds to a concrete physical
feature of the problem. It is nowadays well understood that methods that mimic such underlying geometric structures
have generally a superior dynamical behavior: these methods can in fact be associated
to a nearby dynamical systems with essentially the same dynamics.
See also GeNuIn and
SIG.
Niklas Sæfstøm works on these topics under my supervision.
Nonholonomic mechanical systems are of great interest in robot
technology applications and control, in particular robotic locomotion
and robotic grasping.
Roughly speaking a mechanical system with nonholonomic constraints is
described by a constrained differential equation
such that the constrains are involving the velocity
of the system and not only the positions.
Possible examples are a vertical disk rolling without slipping on a plane, a ball on a
spinning plate (look at some interesting movies here ), the snake-board
(a variant of the skate-board that you can see in the picture above).
The geometry behind these problems is beautifull and non trivial.
The aim of the project is understanding the geometry underlying these problems, design and analyse structure preserving methods for these problems. See also GeNuIn and SIG .
In Norwegian fjords, layers of stratified water with different temperature and salt concentration occur due to ice melting and freshwater supply from rivers. Internal waves are caused by the tide and have a dramatic influence on the ecosystem. We consider convection diffusion PDE models depending on a viscosity parameter $\nu$. The case when the parameter $\nu$ tends to zero is particularly interesting and very challenging from the numerical point of view. In this case the numerical discretizations often lead to phenomena of numerical dispersion. We want to study a new class of integration methods which we believe have a large potential for convection dominated problems. These methods are exponential integrators of Runge-Kutta type. The exact integration of the linear pure convection problems arises as building blocks in the integration method. The resulting methods are semi-Lagrangian.
Bafeh Kingsley Kometa works on this project under my supervision.
We consider problems of
constrained optimization which can be solved
by differential equations on a manifold, like for example the manifold
of $n\times p$ matrices with
orthonormal columns, the Stiefel manifold.
One interesting area of application could be Independent Component Analysis (ICA) in statistical signal processing. A simple example illustrating ICA is the coktail party problem. Suppose we can record the linear mixtures of two signals using two or more microphones in a room. The problem we want to solve is to separate the mixed signals assuming no knowledge of the sources and of the mixing coefficients. For solving this inverse problem a key assumption is that the source signals are statistically independent. See also GeNuIn .
