Wednesdays at 4:15 PM in 384H.
The phaseless reconstruction problem can be stated as follows. Given the magnitudes of a vector coefficients with respect to a linear redundant system (frame), we want to reconstruct the unknown vector. This problem has first occurred in X-ray crystallography starting from the early 20th century. The same nonlinear reconstruction problem shows up in speech processing, particularly in speech recognition. In this talk I present Lipschitz extension results as well as Cramer-Rao Lower Bounds that govern any reconstruction algorithm. In particular we show that the left inverse of the nonlinear analysis map can be extended to the entire measurement space with a small increase in the Lipschitz constant independent of the dimension of the space or the frame redundancy.
Renormalization is a very interesting way of thinking which is distinct from classical techniques of mathematical physics. Many of us, at some time or other, have wondered what it is and tried to learn it, but without success — because its standard applications, quantum field theories and condensed matter physics, are beset with computational complications, while popular expositions don't show how, in a tangible problem, renormalization manages to shake out the critical exponent.
I'd like to present a toy problem where all the difficulties arise, yet which is simple enough that we can really see how the renormalization group resolves them. I'll ensure that anybody who knows calculus can follow everything.
Collective movement is a common yet spectacular manifestation of collective behavior. Despite considerable progress, many of the theoretical principles underlying the emergence of large scale synchronization among moving individuals are still poorly understood. For example, a key question in the study of animal motion is how the details of locomotion, interaction between individuals and the environment contribute to the macroscopic dynamics of the hoard, flock or swarm. The talk will present some of the prevailing models for swarming and collective motion with emphasis on stochastic descriptions. The goal is to identify some generic characteristics regarding the build-up and maintenance of collective order in swarms. In particular, whether order and disorder correspond to different phases, requiring external environmental changes to induce a transition, or rather meta-stable states of the dynamics, suggesting that the emergence of order is kinetic. Different aspects of the phenomenon will be presented, from experiments with locusts to our own attempts towards a statistical physics of collective motion within a simplified network models.
We will discuss general fixed point algorithms for coded-aperture phase retrieval. We will focus on extending von Neumann’s alternating projection algorithm and the classical Douglas-Rachford algorithm to this non-convex setting. In particular, we will present theorems for convergence and uniqueness of fixed point as well as numerical simulations.
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