I have posted below two internal presentations that I delivered that I think may be of general interest. The first is about the transfer learning approach manifold regularization, the second is an overview of PWC-ICA.
I presented a poster on overcomplete representations of EEG signal data and Independent Component Analysis as part of a general meeting with our extended research collaborators:
I presented a talk entitled “Structure-preserving numerical integration and non-coordinate frames” at the AMS Eastern Sectional Meeting on September, 2012 at RIT in Rochester, NY (along with a similar talk the following month at Wake Forest). Below is a link to the .pdf presentation and the original abstract.
In mechanics it may be beneficial to make velocity substitutions that are not tied directly to the configuration coordinates. Such substitutions date back to Euler’s approach to rigid body dynamics, wherein angular velocity is measured against a body frame. Hamel’s formalism may be viewed as a further generalization of Euler’s approach to more general mechanical systems. Variational integrators are algorithms that preserve key mechanical structures and are known to perform well in long term numerical simulations of mechanical systems. Motivated by the effectiveness of Hamel’s formalism in the treatment of mechanical systems with symmetry and velocity constraints, we seek a discretization of Hamel’s equations and will present some recent findings contributing to the area of structure-preserving integrators.
At the 2012 Joint Meetings in Boston I presented a short format talk entitled “Difference equations for long-term simulation of mechanical systems.” Below is a link to the .pdf presentation and the original abstract.
The importance of preservation of various structures of mechanical systems by discrete models has long been acknowledged. It is possible to interpret the dynamics of a mechanical system as a variational problem. Discretizations of variational formulations – as opposed to discretizations of the corresponding differential equations of motion – lead to difference equations that acknowledge these structures and demonstrate good long-term behavior. In this talk we will discuss the extension of this strategy to systems with velocity constraints (such as non-slip conditions on carwheels) using suitable variational principles and show that the resulting difference equations correctly model the constraints.
I gave a talk entitled “The Hamilton-Pontryagin Principle and the Hamel Equations” at the SIAM Southeast Atlantic Section Meeting in March, 2011. Below is a link to the .pdf presentation and the original abstract.
The equations of motion of many mechanical systems can be greatly simplified by utilizing velocity components with respect to a frame that is unrelated to the local configuration coordinates. These Hamel equations are equivalent to Hamilton’s principle subject to constrained variations in the system’s velocity phase. In this talk, we show that the Hamel equations can be derived by taking unconstrained variations in a concatenated momentum-velocity phase space. We expect this constraint-free derivation to be useful in the study of variational integrators.
I presented the linked poster at a couple of conferences, once in Orlando at UCF in the Fall of 2009 and an additional time in Raleigh during the Spring of 2010.