Applied Math Seminar

Polarization Based Phase Retrieval for Time-Frequency Structured Measurements

04/30/2015 - 2:30pm
04/30/2015 - 3:30pm
Speaker: 
Palina Salanevich
Abstract: 

 In many areas of imaging science, such as diffraction imaging, astronomical imaging, microscopy, etc., optical detectors can often only record the squared modulus of the Fraunhofer or Fresnel diff raction pattern of the radiation that is scattered from an object. In such setting, it is not possible to measure the phase of the optical wave reaching the detector. So, it is needed to reconstruct a signal from intensity measurements only. This problem is called phase retrieval.

We are going to consider the case when the measurement frame is a Gabor frame, that is, the case of time-frequency structured measurements. The main motivation is that in this case, the frame coefficients are of the form of masked Fourier coefficients, where the masks are time shifts of the Gabor window. This makes measurements meaningful for applications, but at the same time preserves the flexibility of the frame-theoretic approach. The most efficient existing algorithms, such as PhaseLift, work with randomly generated Gaussian frames. I am going to present the recovery algorithm with a sufficiently small number of measurements required, which is working with time-frequency structured measurements. The algorithm is based on the idea of polarization, first proposed by Alexeev, Bandeira, Fickus and Mixon.

 

 

Where: 
Roberts North (RN) 103

Broad Band Solitons in a Periodic and Nonlinear Maxwell System

03/20/2015 - 3:00pm
03/20/2015 - 4:00pm
Speaker: 
Dmitry Pelinovsky (McMaster University)
Abstract: 

We consider the one-dimensional Maxwell equations with low contrast periodic linear refractive index and weak Kerr nonlinearity. Wave packet initial conditions with a single carrier frequency excite infinitely many resonances. On large but finite time-scales, the coupled evolution of backward and forward waves is governed by nonlocal equations of resonant nonlinear geometrical optics. For the special class of solutions which are periodic in the fast phase, these equations are equivalent to an infinite system of nonlinear coupled mode equations. Numerical studies support the existence of long-lived spatially localized coherent structures, consisting of a slowly varying envelope of a train of carrier shocks. This presentation is based on the joint work with G. Simpson and M. Weinstein.

Where: 
CGU Math South (710 N. College Ave)

Geometric methods for graph partitioning

05/04/2015 - 12:00pm
05/04/2015 - 1:00pm
Speaker: 
Braxton Osting (University of Utah)
Abstract: 

Several geometric methods for graph partitioning have been introduced in the past few years, with wide applications in clustering, community detection, and image analysis. These methods, which I'll review, are built on graph-based analogues of total variation, motion by mean curvature, the Ginzburg-Landau functional, and the Merriman-Bence-Osher threshold dynamics. In this talk, I'll discuss a new graph partitioning method where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. The resulting eigenvalue optimization problem can be solved by a rearrangement algorithm, which we show to converge in a finite number of iterations to a local minimum of a relaxed objective function. The method compares well to state-of-the-art approaches when applied to clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations. The model has a semi-supervised extension and provides natural representatives for the clusters as well.

Where: 
Kravis 100

Local convergence of an algorithm for subspace identification with missing data

02/17/2015 - 10:00am
02/17/2015 - 11:00am
Speaker: 
Laura Balzano (U of Michigan)
Abstract: 

Low-dimensional linear subspace approximations to high-dimensional data find application in a great variety of applications where missing data are the norm, not only because of errors and failures in data collection, but because it may be impossible to collect and process all the desired measurements.

In this talk, I will describe recent results on estimating subspace projections from incomplete data. I will discuss the convergence guarantees and performance of the algorithm GROUSE (Grassmannian Rank-One Update Subspace Estimation), a subspace tracking algorithm that performs gradient descent on the Grassmannian. I will also discuss the relationship of GROUSE with an incremental SVD algorithm, and show results of GROUSE applied to problems in computer vision.

Where: 
RN 104
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Graph representatives of positroid strata

03/30/2015 - 4:15pm
03/30/2015 - 5:15pm
Speaker: 
Puck Rombach (UCLA)
Abstract: 

This talk will be very accessible (including to grad students) and will involve juggling. The positroid stratification, studied by many authors, is a coarsening of the matroid stratification of the Grassmannian. Each graph (with orientation and edge-ordering) gives a point in the Grassmannian; for a matroid stratum to contain such a point is a well-known forbidden minor condition on the matroid. We show that, by contrast, every positroid stratum contains a graphical representative; indeed, one can choose the graph to be planar. This is despite the fact that the matroid stratum dense in the positroid stratum does not typically contain such a representative (“positroids are not graphic matroids”). Joint work with Allen Knutson.

Where: 
Kravis 102

LOCALIZATION OF COMPACT INVARIANT SETS OF NONLINEAR SYSTEMS AND GLOBAL STABILITY IN THE ANALYSIS OF CANCER TUMOR GROWTH MODELS

05/11/2015 - 12:00pm
05/11/2015 - 1:00pm
Speaker: 
KonstantinEv. Starkov (Instituto Politecnico Nacional CITEDI, Mexico)
Abstract: 

The localization method of compact invariant sets based on a solution of the conditional extremum problem is  applied to analysis of global dynamics for some bladder tumor growth models. The main attention is attracted to finding ultimate upper and lower bounds of cells populations, analysis of the dissipativity in the Levinson sense property and derivation of global stability conditions of the tumor-free equilibrium point. Briefly applications of this method to the analysis of some other classes of nonlinear systems including Hamiltonian are discussed.

 

Where: 
Kravis 100

Voronoi set approximation

03/09/2015 - 12:00pm
03/09/2015 - 1:00pm
Speaker: 
Raphael Lachieze-Rey (Paris Descartes University)
Abstract: 

Set approximation consists in the reconstruction of an unknown bounded set K, based on a finite random sampling in the region where K is supposed to lie. We are concerned here with a specific procedure called "Voronoi approximation", where one takes the union of all Voronoi cells whose centers lie in K. We will discuss the quality of this approximation when the number of random sampling points goes to infinity. We will in particular present Berry-Esseen bound on the volume approximation, and an a.s. convergence result for the Hausdorff distance. We are also interested in the minimal regularity assumptions required on the set K, and will show that the results even apply to sets with a possibly fractal boundary, such as the Von Koch flake.

Where: 
Kravis 100

Compressed Modes and Localized Density Matrices

04/27/2015 - 12:00pm
04/27/2015 - 1:00pm
Speaker: 
Rongjie Lai (Rensselaer Polytechnic Institute)
Abstract: 

 $\ell_1$ regularization for sparsity has played important role in recent developments in many fields including signal processing, statistics, optimization. The concept of sparsity is usually for the coefficients (i.e., only a small set of coefficients are nonzero) in a well-chosen set of modes (e.g. a basis or dictionary) for representation of the corresponding vectors or functions. Our recent work investigate a new use of sparsity-promoting techniques to produce “compressed modes" - modes that are sparse and localized in space - for efficient solutions of constrained variational problems in mathematics and physics.  I first will discuss L1 regularized variational Schrodinger equations for creating spatially localized modes and orthonormal basis, which can efficiently represent localized functions.  In addition, I will also discuss our recent work on localized density matrices and their linear scaling algorithms.

 

Where: 
Kravis 100

Atomistic Nanoscale Device Simulation and Its Compact Circuit Modeling

03/02/2015 - 12:00pm
03/02/2015 - 1:00pm
Speaker: 
Shigeyasu Uno (Ritsumeikan University)
Abstract: 
Metal-oxide-semiconductor field-effect-transistors (MOSFETs) are known as key semiconductor devices used in processors and memories. The size of MOSFETs has continuously been shrunk since they are commercialized several decades ago, and now the minimum device size is as small as several 10nm. In such small devices, various interesting physics emerges, such as quantum-mechanical effects, atomistic discreteness, and quasi-ballistic electron transport. Device and circuit simulations of such nanoscale MOSFETs require new modeling frameworks, and we have been working on developing them.
In this talk, I will mainly talk about compact circuit model of such nanoscale MOSFETs and its use in circuit simulation. I will also touch upon numerical device simulations based on atomistic band structures and quantum electron transport in non-equilibrium green’s function formalism. The major results are outcomes of collaboration among several universities in CREST project, Japanese government research funding scheme.
Where: 
Kravis 100

Evolution of dispersal in heterogeneous landscape

04/06/2015 - 12:00pm
04/06/2015 - 1:00pm
Speaker: 
Yuan Lou (The Ohio State University)
Abstract: 

From habitat degradation and climate change to spatial spread of invasive species, dispersal plays a central role in determining how organisms cope with environment. How should organisms disperse "optimally" in heterogeneous environments? I will discuss some recent development on the evolution of dispersal, focusing on evolutionarily stable dispersal strategies in PDE models.

Where: 
Kravis 100
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