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. In particular classical Runge-Kutta methods will be first applied to the problems and then more specific geometric integrators will be also used. A comparison of the performance of the methods will be part of the presented results. See also NHreport1.pdf for some preliminary work on this topic. The student can choose if to explore the more teoretical or the more practical aspects of the considered problems.
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.
For more information see also the presentation edinburgh.pdf
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.
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.
