K.T. Chen considered sequencies of iterated integrals of smooth or piecewise smooth paths on manifolds and studied their geometric importance. His reluts have relevance to the calculus of variations and were used by Fleiss in the context of contol theory, and Platen in stochastic differential equations. Iterated integrals can be used to obtain a Taylor expansion of arbitrary order for the solution of such equations.
We aim here at studying the basics of rough paths theory and their connection to the study of B-series and Lie-Butcher series for numerical integrators. The project can explore applications of the theory to the solution of stochastic differential equations as well as algebraic aspects of the considered series.
If we are able to solve this problem, we might be able to modify the robot and eventually, by similar techniques, simulate its capabilities as well as uncover any possible instabilities of the mechanical system. See here for more information.
There is a range of problems one could study in master projects, and good opportunities for starting a PhD after the master. For example geometric numerical integration methods for the equations of motion of problems of rigid body and rod dynamics could be studied, analyzed and implemented. The methods could be applied to simulation and control of pipe-lay operations.
In practice one should learn about one or several of the following: rigid-body dynamics, basics of differential geometry and Lie groups (especially SO(3)), geometric numerical methods for canonical and non-canonical Hamiltonian systems, splitting methods, geometric methods for PDEs, Jacobi elliptic functions and quaternions.
More information can be found in the page of the GeNuIn project (see also under publications) and in the page of the Special Interest Group in Motion Control by Geometric methods. Einar Gustafsson is presently writing his master thesis in this field.
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. In this project the numerical simulation of some simple noholonomic mechanical systems will be considered. 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 side picture). The geometry behind these problems is beautifull and non trivial.
The aim of the project is understanding the basic theoretical features of nonholonomically constrained systems, illustrate them via numerical simulation, and discuss which numerical approaches are best suited for such problems. So far SPARK Runge-Kutta methods and Non-holonomic variational integrators have been tested. These methods perform well and correspond to a discretization of different variational principles. Energy preservation under discretization more than symplecticity seams to be of importance in these problems.
The student can choose to explore either the more teoretical or the more practical aspects of the considered problems. Dag Frodhe Evensberget and Sindre Hilden have written their master thesis on this subject. For more information on this topic do not hesitate to contact me.
The brain is a massively parallel information processing system. Some of the goals of neural networks theory are: achieving efficient use of machines in tasks currently solved by humans, improve computational capabilities taking the brain as a model, understand how the brain works. The tansmission of signals between neurons and the alteration of the strenght of the connection between neurons characterize the learning process. Engineers deviced simple models of neurons where $n$ inputs are processed in a neuron end $p$ outputs are generated. Combining several neurons one obtains a network. The rules for learning are translated into differential equations for the network.
One succesfull area of application of Neural Networks is 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. This can be done using artificial Neural Networks and designing suitable learning equations. Typically the problem can be formulated as a constrained optimization problem which can be solved by differential equations on the manifold of $n\times p$ matrices with orthonormal columns, the Stiefel manifold.
Optimization on Riemannian manifolds can be used also in the analysis of shapes and is of relevance in problems of face and object recognition. Eivind Fonn wrote his diploma thesis on this subject and received the NRS prize for the best master thesis in Mathematics and ICT related subjects at UiO and NTNU.
The project will consider an introduction to the theory of artificial neural Networks and Independent Component Analysis. Signal separation techniques will be implemented using numerical integration methods on manifolds.
Alternatively one could look at some other problem where optimization on manifolds is relevant. In these cases the study of the manifold is the first step of the process. The optimization tools are Riemannian and sub-Riemannian metrics for the definition of the objective functions, and the computation of geodesics to solve the optimization problems.
Thesis supervised by Olivier Verdier
For these master projects Olivier will be the actual supervisor and I will act only as the formal supervisor. Olivier's projects .
Simulation of the double pendulum equations.
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.
In collaboration with the microelectonics company MAGWEL we consider the iterative solution of large linear algebraic systems arising from the numerical discretization of Maxwell's equations. The systems are sparse and potentially very large (up to 600000 unknowns). The students will work on finding good preconditioning techniques for these systems to be implemented on parallel architectures. The performance of the proposed methods will be compared to the approach presently used by MAGWELL.
