Applied Math Seminar

CFD Simulations and Particulate Deposition Before and After Intervention in Patient with Tracheal Stenosis

04/25/2016 - 4:15pm
04/25/2016 - 5:15pm
Speaker: 
Shahab Taherian (California State University, Long Beach)
Abstract: 

The existence of obstructions such as tracheal stenosis has major impacts on respiratory functions. Therapeutic effectiveness of inhaled medications is influenced by tracheal stenosis, and particle transport and deposition pattern are modified. The majority of studies have focused on obstructions in branches of the airways, where air is diverted to the other branches to meet the needed oxygen intake. In this study we have investigated the effects of trachea with and without stenosis/obstruction on particle depositions and air flow in a human respiratory system, using patient-specific Computational Fluid Dynamics (CFD) simulations and CT-scans.

Where: 
Emmy Noether Room Millikan 1021 Pomona College

Viscosity Solutions of Path Dependent PDEs

04/11/2016 - 4:15pm
04/11/2016 - 5:15pm
Speaker: 
Jianfeng Zhang (USC)
Abstract: 

Abstract: Path dependent PDEs considers continuous paths as its variable. It is a convenient tool for stochastic optimization/games in non-Markovian setting, and has natural applications on non-Markovian financial models with drift and/or volatility uncertainty. For example, a martingale can be viewed as a solution to a path dependent heat equation, and we are particularly interested in path dependent HJB equations and Isaacs equations. In path dependent case, even a heat equation typically does not have a classical solution, where smoothness is defined through Dupire's functional Ito calculus, so a viscosity theory is desirable. There are two major difficulties in the project: (i) the state space of (continuous paths) is not locally compact, and thus one cannot apply many tools in standard viscosity theory; (ii) fully nonlinear PPDEs involve a nonlinear expectation under which the dominated convergence theorem fails. In this talk, we will motivate our definition of viscosity solutions and give an overview of the recent developments of the theory.

Where: 
Emmy Noether Room Millikan 1021 Pomona College

Adaptive Finite Element Method for PDEs on Surfaces

02/22/2016 - 4:15pm
02/22/2016 - 5:15pm
Speaker: 
Ryan S. Szypowski (Cal Poly, Pomona)
Abstract: 

The finite element method is a powerful technique for approximating
solutions to partial differential equations (PDEs) that is based on
rich theory and is efficiently implementable. When used in an adaptive
fashion, the method is provably convergent for a wide array of
problems. The recently developed Finite Element Exterior Calculus
formalism allows the method to be applied to problems with geometric
content. This talk will introduce the basics of this formalism,
specifically in the context of PDEs on surfaces, and provide some
recent theoretical and numerical results.

Where: 
Emmy Noether Room Millikan 1021 Pomona College

Verblunsky Coefficients (and Related Ideas)

04/04/2016 - 4:15pm
04/04/2016 - 5:15pm
Speaker: 
Angel Chavez (Pomona College)
Abstract: 

Verblunsky coefficients provide a remarkable correspondence between sequences of points in the unit disk and ``non-trivial” probability measures on the unit circle. In this talk, we’ll give an overview of this Verblunksy correspondence and then discuss progress on two projects pertaining to Verblunsky coefficients (and related ideas).

Where: 
Emmy Noether Room Millikan 1021 Pomona College

A Strong Law and a Law of the Single Logarithm for Arrays of Row-wise Independent Random Variables

02/01/2016 - 4:15pm
02/01/2016 - 5:15pm
Speaker: 
Tien-Chung Hu (Department of Mathematics, National Tsing Hua University )
Abstract: 

 See the attachment

Where: 
Millikan 1021 Pomona College

An estimate for the probability of heads on a coin whose relative error is independent of the true value

02/29/2016 - 4:15pm
02/29/2016 - 5:15pm
Speaker: 
Mark Huber (Claremont McKenna College)
Abstract: 

Consider a coin with an unknown probability $ p $ of heads that can be flipped as many times as needed. In this talk I will present a new estimate $ \hat p $ for $ p $ such that the relative error $ (\hat p/p) - 1 $ has a distribution that is independent of $ p $. This has applications in several Monte Carlo algorithms, including using acceptance/rejection to give approximations of high dimensional integrals and sums. In addition, this idea can also be used to obtain an estimate with similar properties for the mean of a sequence of independent, identically distributed Poisson random variables.

Where: 
Millikan 1021 Pomona College

Multiphase flow modeling of cell adhesion and migration

03/07/2016 - 4:15pm
03/07/2016 - 5:15pm
Speaker: 
Damir Khismatullin (Biomedical Engineering, Tulane U.)
Abstract: 

One of the fundamental properties of living cells is their ability to migrate from one region of space to another in response to specific chemical stimuli. Cell migration often involves adhesion to the surrounding tissue, chemoattractant-mediated intracellular signaling, and intracellular force generation. We have developed state-of-the-art three-dimensional computational algorithms, known as VECAM and VECAM-Active, to simulate cell-substrate adhesion and active, amoeboid migration of motile cells. These algorithms are based on the multiphase flow approach and account for cell and substrate deformability, rheological properties, multiple cellular compartments, receptor-ligand binding, transport of chemical activators (chemokines, cytokines), intracellular force generation due to actin polymerization, and physiologic shear flow conditions. In this talk, I will first present recent experimental data of my laboratory on circulating cell adhesion to activated vascular endothelium. Then I will show VECAM data on passive migration of cells in microchannels with different geometry (straight, Y-junction bifurcation, cross-flow, grooves and pillars), deformable cell adhesion to a receptor-coated substrate, and chemotactic and haptotactic migration of cells in a microchannel.

Where: 
Emmy Noether Room Millikan 1021 Pomona College

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
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