Applied Math Seminar

Data-driven signal decomposition into primary building blocks

02/08/2016 - 4:15pm
02/08/2016 - 5:15pm
Speaker: 
Charles K. Chui (Department of Statistics, Stanford University)
Abstract: 

The strategy of divide-and-conquer applies to just about all scientific and engineering disciplines for theoretical and algorithmic development as well as experimental implementations for various applications. In mathematics, perhaps one of the most exciting theoretical development in this direction is the notion of “atomic decomposition” for the Hardy spaces $H^p(\R)$ with $0<p\le1$, introduced by Raphy Coifman in a 1974 paper, which contributed to motivate his joint work with Yves Meyer and Elias Stein, published some 10 years later, on the introduction and characterization of the so-called Tent spaces. This significant piece of work has important applications to the unification and simplification of the basic techniques in harmonic analysis. Furthermore, the atomic decomposition of these and other function spaces, contributed by others, has profound impact to the advancement of both harmonic and functional analyses over the decades of the 80’s and 90’s. An important property of Coifman’s atoms for $H^p(\R)$, with $0<p\le1$, is their vanishing moments of order up to $1/p$, leading to the introduction of wavelets and the rapid advances of wavelet analysis and algorithmic development, with applications to most engineering and physical science disciplines for a duration of over two decades.

With the popularity of wavelets, several other families of wavelet-like basis functions were introduced, including: wavelet packets, localized cosines, chirplets, warplets, and mulit-scale Gabor dictionaries. Using this large collection of basis functions as “atoms” to compile a desirable dictionary, the powerful mathematical tool of “basis pursuit”, introduced and studied in some depth by Scott Chen, David Donoho, and Michael Saunders in their popular 1999 SIAM J. Scientific Computing paper, provides a more attractive alternative of the standard iterative computational schemes based on “matching pursuit”. On the other hand, it is very difficult, even if feasible at all, to compile an effective dictionary for sufficiently accurate decomposition of real-world signals in general. In this regard, the basis pursuit approach was recently modified by Thomas Hou and Zuoqiang Shi to “nonlinear basis pursuit”, using a nonlinear optimization scheme. Unfortunately, this approach requires appropriate guesses of initial phase functions, and perhaps the effectiveness of the guess might also depend on the (unknown) number of atoms.

In a recent joint paper with Hrushikesh Mhaskar and my student, Maria van der Walt, of Vanderbilt University, we have initiated a study of the construction of atoms for signal decomposition directly from the input signal itself. The objective of this seminar is to discuss the background of our approach, compare our results with the state-of-the-arts, and conclude the presentation with a brief discussion our computational scheme to signal extrapolation.

 

Where: 
Emmy Noether Room Millikan 1021 Pomona College

The fluid dynamics of jellyfish swimming and feeding

02/15/2016 - 4:15pm
02/15/2016 - 5:15pm
Speaker: 
Laura Miller (UNC)
Abstract: 

The jellyfish has been the subject of numerous mathematical and physical studies ranging from the discovery of reentry phenomenon in electrophysiology to the development of axisymmetric methods for solving fluid-structure interaction problems. In this presentation, we develop and test mathematical models describing the pulsing dynamics and the resulting fluid flow generated by the benthic upside down jellyfish, Cassiopea spp., and the pelagic moon jellyfish, Aurelia spp. The kinematics of contraction and distributions of pulse frequencies were obtained from videos and used as inputs into numerical simulations. Particle image velocimetry was used to obtain spatially and temporally resolved flow fields experimentally. The immersed boundary method was then used to solve the fluid-structure interaction problem and explore how changes in morphology and pulsing dynamics alter the resulting fluid flow. For Cassiopea, significant mixing occurs around and directly above the oral arms and secondary mouths. We found good agreement between the numerical simulations and experiments, suggesting that the presence of porous oral arms induce net horizontal flow towards the bell and mixing. For Aurelia, maximum swim speeds are generated when the elastic bell is resonating at its natural frequency. Alternating vortex rings can also enhance swimming speed and efficiency.

Where: 
Emmy Noether Room Millikan 1021 Pomona College

Graph Expansion and its Relationship with Probability and Geometry

12/07/2015 - 4:15pm
12/07/2015 - 5:15pm
Speaker: 
Angelica Gonzalez (University of Arizona)
Abstract: 

Thinking of a graph as a network, the expansion constant measures how efficient a graph is
with respect to optimization of cost, connectivity, and robustness. The expansion constant of a graph
measures the sparsity (in terms of the number of edges, relative to the number of vertices) while still
maintaining connectivity of a graph. In this talk we explore the notion of the expansion constant of
a graph and it's relationship to the spectrum of the adjacency matrix of a graph. This will lead to
a discussion of how some geometric and probabilistic techniques are useful in the study of expansion.
We will conclude by investigating some of these notions for a specifi c class of 3-regular graphs.

Where: 
Emmy Noether Room, Millikan 1021, Pomona College

Optimization problems on planar lattice transmitter networks

10/26/2015 - 4:15pm
10/26/2015 - 5:15pm
Speaker: 
Lenny Fukshansky (CMC)
Abstract: 

Consider a network of radio transmitters of equal power placed at points of a two-dimensional lattice. On the one hand, we would like to choose a lattice that would maximize the coverage area of our network; on the other, we want to maximize the signal-to-noise ratio. We present some results and conjectures in the direction of these optimization problems and their relation to each other on some particularly interesting classes of planar lattices, as well as discuss some counting results on the number of possible such lattice networks of fixed determinant. This work came out of a Fletcher Jones Fellowship project in 2011.

Where: 
Emmy Noether Room Millikan 1021 Pomona College

Optimal control for chemotherapy scheduling

09/28/2015 - 4:00pm
09/28/2015 - 5:00pm
Speaker: 
Minaya Villasana (Universidad Simón Bolívar)
Abstract: 

In this talk some advances on the optimal control model formulation and resolution for chemotherapy scheduling is presented.  Firstly, a DDE model is introduced for tumor progression with immune response upon which a suitable optimal control problem formulation is devised and several heuristic algorithms are evaluated.  A second model that incorporates two drugs: cytotoxic and cytostatic is presented.  We show numerically that the optimal scheduling on this new model corresponds to an asynchronous delivery. 

Where: 
Emmy Noether Room, Millikan 1021, Pomona College

Topological data analysis for investigation of dynamics and networks

11/09/2015 - 4:15pm
11/09/2015 - 5:15pm
Speaker: 
Heather Harrington (Oxford University)
Abstract: 

Persistent homology (PH) is a technique in topological data analysis that allows one to examine features in data across multiple scales in a robust and mathematically principled manner, and it is being applied to an increasingly diverse set of applications. We investigate applications of PH to dynamics and networks, focusing on two settings: dynamics {\em on} a network and dynamics {\em of} a network.

Dynamics on a network: a contagion spreading on a network is influenced by the spatial embeddedness of the network. In modern day, contagions can spread as a wave and through the appearance of new clusters via long-range edges, such as international flights. We study contagions by constructing Œcontagion maps¹ that use multiple contagions on a network to map the nodes as a point cloud. By analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modelling, forecast, and control of spreading processes.

Dynamics of a network: one can construct static graph snapshots to represent a network that changes in time (e.g. edges are added/removed). We show that partitionings of a network of random-graph ensembles into snapshots using existing methods often lack meaningful temporal structure that corresponds to features of the underlying system. We apply persistent homology to track the topology of a network over time and distinguish important temporal features from trivial ones. We define two types of topological spaces derived from temporal networks and use persistent homology to generate a temporal profile for a network. We show that the methods we apply from computational topology can distinguish temporal distributions and  provide a high-level summary of temporal structure.

Together, these two investigations illustrate that persistent homology can be very illuminating in the study of networks and their applications.

 

 

Where: 
Emmy Noether Room, Millikan 1021, Pomona College

Incorporating Sustainability into Mathematics Courses

11/23/2015 - 4:15pm
11/23/2015 - 5:15pm
Speaker: 
Victor J. Donnay (Bryn Mawr College)
Abstract: 

Abstract: How to link the math with the real world? Professor Donnay will discuss his experiences of incorporating issues of environmental sustainability into his math teaching.  He will present a variety of approaches ranging from easy to adapt extensions of standard homework problems to elaborate service learning projects. He will share some of the educational resources that he helped collect as chair of the planning committee for Mathematics Awareness Month 2013: the Mathematics of Sustainability, as well as his TED Ed video on Tipping Points and Climate Change. He has used these approaches in a variety of courses including Calculus, Differential Equations, Mathematical Modeling, and a Senior Seminar.

In addition to students and faculty, secondary math and science teachers are invited to attend.
Where: 
Emmy Noether Room, Millikan 1021, Pomona College

Constrained Adaptive Sensing

10/12/2015 - 4:15pm
10/12/2015 - 5:15pm
Speaker: 
Tina Woolf (CGU)
Abstract: 

Traditional signal processing schemes sample signals at a high rate and immediately discard most of the information during the compression process. Compressed sensing is a new field that improves this by directly sensing the signal in compressed form using few nonadaptive, linear measurements. Adaptive sensing, which allows the selection of the next measurement based on previous observations, significantly improves signal recovery when arbitrary linear measurements can be constructed. However, in practice, the types of measurements that can be acquired are limited. In this talk, we will discuss recent results on the limitations and advantages of adaptive sensing when the measurements are constrained to belong to a finite set of allowable measurement vectors.

Where: 
Emmy Noether Room (Pomona), Millikan 1021

Modeling Electrical Capacitance Tomography with Complete Electrode Model

10/05/2015 - 4:15pm
10/05/2015 - 5:15pm
Speaker: 
Weifu Fang (Mathematics and Statistics, Wright State University)
Abstract: 

We will discuss the modeling of electrical capacitance tomography with the use of different electrode models, in comparison with the study of electrical impedance tomography.

 

Where: 
Emmy Noether Room, Millikan 1021, Pomona College

On a "Lethargic" Theorem

09/21/2015 - 4:00pm
09/21/2015 - 5:00pm
Speaker: 
Asuman Aksoy (Claremont McKenna College)
Abstract: 

One of the most important theorems used in constructive theory of functions is called Berstein's "lethary" theorem [3]. The theorem states that if $ \{d_i\}  $ is a sequence of nonnegative numbers with $ \lim d_i =0 $ then there exists a function $ f\in C[0,1] $ such that dist$ (f, P_i) = d_i $ for $ i=0,1,2,3,\dots $, where C[0,1] denotes the Banach space of all continuous, real-valued functions defined on the interval $ [0,1] $ with supremum norm, and $ P_i $ denotes the space of all polynomials of degree $  \leq i $. In this talk, after examining the developments in this theory [1], we present a lethargic theorem for Fr ́echet spaces [2]. This is joint work with G. Lewicki.

References

[1] J. Almira and A. G. Aksoy On Shapiro’s Lethargy Theorem and Some Appli- cations, Jean. J. Approx. 6(1), 87 − 116, 2014.

[2] A. G. Aksoy and G. Lewicki Bernstein’s Lethargy Theorem in Fr ́echet spaces, arXiv:1503.06190.

[3] S. N. Bernstein, Collected Works, II Moskow,: Akad Nauk SSR. 1954.

Where: 
Emmy Noether Room, Millikan 1021, Pomona College
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