Arthur Szlam : A Total Variation-based Graph Clustering Algorithm for Cheeger Ratio Cuts
- Undergraduate Seminars ( 261 Views )I will discuss a continuous relaxation of the Cheeger cut problem on a weighted graph, and show how the relaxation is actually equivalent to the original problem. Then I will introduce an algorithm which experimentally is very efficient at approximating the solution to this problem on some clustering benchmarks. I will also give a heuristic variant of the algorithm which is faster but often gives just as accurate clustering results. This is joint work with Xavier Bresson, inspired by recent papers of Buhler and Hein, and Goldstein and Osher, and by an older paper of Strang.
Elliott Wolf & Alex Woolf : CONVEX-OPTIMIZING THE POWER GRID
- Undergraduate Seminars ( 251 Views )The addition of renewable energy sources, whose power production cannot be scheduled, has created increasing gaps between instantaneous electricity supply and electricity demand. Sometimes the grid is oversupplied with energy, requiring zero-marginal-cost sources of power to be shut or energy to be bled off of the grid. Other times there is insufficient electricity, requiring high-marginal-cost sources of electricity to be switched on or consumers to curtail their demand. The current state of the grid has led various utilities and power consumers deploy capital-intensive energy storage, such as lithium-ion batteries, to better-match grid supply with grid demand. We present a method to add large-scale energy storage to the power grid using only sensors, software modifications to the control systems of large industrial refrigeration systems, and mathematical optimization. Our talk will address the required instrumentation, the physics necessary to understand applicable thermal constraints, and numerical methods used to determine a mathematically optimal-discharge schedule. We further discuss the economics of the US power grid, "war stories"of doing complex mathematics in a large industrial setting and the effects of various Federal Energy Regulatory Commission and California Public Utility Commission on our efforts.
Mainak Patel : The Essential Role of Phase Delayed Inhibition in Decoding Synchronized Oscillations within the Brain
- Undergraduate Seminars ( 242 Views )The widespread presence of synchronized neuronal oscillations within the brain suggests that a mechanism must exist that is capable of decoding such activity. Two realistic designs for such a decoder include: 1) a read-out neuron with a high spike threshold, or 2) a phase-delayed inhibition network motif. Despite requiring a more elaborate network architecture, phase-delayed inhibition has been observed in multiple systems, suggesting that it may provide inherent advantages over simply imposing a high spike threshold. We use a computational and mathematical approach to investigate the efficacy of the phase-delayed inhibition motif in detecting synchronized oscillations, showing that phase-delayed inhibition is capable of detecting synchrony far more robustly than a high spike threshold detector. Furthermore, we show that in a system with noisy encoders where stimuli are encoded through synchrony, phase-delayed inhibition enables the creation of a decoder that can respond both reliably and specifically to a stimulus, while a high spike threshold does not.
Jim Nolen : Bumps in the road: stability and fluctuations for traveling waves in an inhomogeneous medium
- Undergraduate Seminars ( 236 Views )If a partial differential equation has coefficients that vary with respect to the independent (spatial) variables, how do the fluctuations in the coefficients effect the solution? In particular, if these fluctuations have a statistical structure, can anything thing be said about the statistical behavior of the solutions? I'll consider these questions in the context of a scalar reaction diffusion equation. Without the variable coefficients, the equation admits stable traveling wave solutions. It turns out that the stability of wave-like solutions persists in a heterogeneous environment, and this fact can be used to derive a central limit theorem for the wave when the environment has a certain statistical structure.
I'll try to explain two interesting mathematical issues: First, how can one prove stability of the wave-like solution in this general setting, since spectral techniques don't seem applicable? Second, how can one use the structure of the problem to say something about how randomness in the environment effects the solution?