Applied Math Seminar

Engineering Very Long-Lived States

11/02/2011 - 1:15pm
11/02/2011 - 2:15pm
Speaker: 
Braxton Osting, UCLA
Abstract: 

In many optical and quantum systems it is desirable to spatially confine energy in a particular mode for a long period of time. I'll discuss the mathematics of energy-conserving, spatially-extended systems and present analytical and computational results on optimal energy confining structures.

 

Where: 
Davidson at CMC

Finding Structure with Randomness: Probabilistic Algorithms for Constructing Low-Rank Matrix Decompositions

10/26/2011 - 1:00pm
10/26/2011 - 2:00pm
Speaker: 
Joel Tropp, California Institute of Technology
Abstract: 


Finding structure with randomness: Probabilistic algorithms for
constructing low-rank matrix decompositions

Computer scientists have long known that randomness can be used to
improve the performance of algorithms. A familiar application is the
process of dimension reduction, in which a random map transports data
from a high-dimensional space to a lower-dimensional space while
approximately preserving some geometric properties. By operating with
the compact representation of the data, it is theoretically possible
to produce approximate solutions to certain large problems very
efficiently.

Recently, it has been observed that dimension reduction has powerful
applications in numerical linear algebra and numerical analysis. This
talk provides a high-level introduction to randomized methods for
computing standard matrix approximations, and it summarizes a new
analysis that offers (nearly) optimal bounds on the performance of
these methods. In practice, the techniques are so effective that they
compete with—or even outperform—classical algorithms. Since matrix
approximations play a ubiquitous role in areas ranging from
information processing to scientific computing, it seems certain that
randomized algorithms will eventually supplant the standard methods in
some application domains.

Joint work with Gunnar Martinsson and Nathan Halko. The paper is available at

http://users.cms.caltech.edu/~jtropp/papers/HMT11-Finding-Structure-SIRE...

 

Where: 
Bauer Court (BC) 24, in CMC

A new approach to regularity and singularity questions for a class of non-linear evolutionary PDEs (eg, 3-D Navier-Stokes eqns)

03/09/2010 - 4:15pm
03/09/2010 - 5:15pm
Speaker: 
Saleh Tanveer (The Ohio State University)
Abstract: 

(Joint work with Ovidiu Costin, G. Luo.)

Abstract:
-------------
We consider a new approach to a class of evolutionary PDEs where question of global existence or lack of it is tied to the asymptotics of solution to a non-linear integral equation in a dual variable whose solution has been shown to exist a priori. This integral equation approach is inspired by Borel summation of a formally divergent series for small time, but has general applicability and is not limited to analytic initial data.

In this approach, there is no blow-up in the variable p, which is dual to 1/t or some power 1/t^n; solutions are known to be smooth in p and exist globally for p in R+. Exponential growth in p, for different choice of n, signifies finite time singularity. On the other hand, sub-exponential growth implies global existence.

Further, unlike PDE problems where global existence is uncertain, a discretized Galerkin approximation to the associated integral equation has controlled errors. Further, known integral solution for p in [0, p_0], numerically or otherwise, gives sharper analytic bounds on the exponents in p and hence better estimate on the existence time for the associated PDE.

We will also discuss particular results for 3-D Navier-Stokes and discuss ways in which this method may be relevant to numerical studies of finite time blow-up problems.

Where: 
Third Floor - Sprague Building

Computational Modeling of the Shoot Apical Meristem

04/23/2008 - 12:00pm
04/23/2008 - 1:00pm
Speaker: 
Dr. Bruce Shapiro (Caltech, JPL & CSU Northridge)
Abstract: 

The shoot apical meristem (SAM) is a dome-shaped collection of cells at the apex
of growing plants from which all above-ground tissue ultimately derives. In Arabidopsis
thaliana (thale cress), a small flowering weed of the Brassicaceae family (related
to mustard and cabbage), the SAM typically contains some three to five hundred
cells that range from five to ten microns in diameter. These cells are organized into
several distinct zones that maintain their topological and functional relationships
throughout the life of the plant. As the plant grows, organs (primordia) form on its
surface flanks in a phyllotactic pattern that develop into new shoots, leaves, and
flowers. The central region contains pluripotent stem cells that continue to divide
and differentiate into mature tissue throughout the life of the plant. In the computable
plant project we observe several cell type-specific markers for growth and differentiation
in live Arabidopsis plants with a dedicated confocal laser scanning microscope. These
markers are affixed to various gene products or promoter regions using green fluorescent
protein (GFP) variants that flouresce when they are illuminated within the microscope by
a laser. This allows us to observe various meristem and floral primordial features, such
as membranes and nuclei, and to track specific cell lineages over time. By using
mathematical and computational models of these spatiotemporal expression patterns,
we can infer how primordial cells are progressively specified and organs develop.
The talk will survey the modeling techniques and tools used and the modeling results
produced in this project.

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

Combinatorial Optimization, Phase Transitions, and the Peculiar Case of Graph Bisection

04/18/2008 - 3:00pm
04/18/2008 - 4:00pm
Speaker: 
Dr. Allon Percus (Los Alamos National Laboratory)
Abstract: 

Combinatorial optimization problems, where one must minimize an
objective function of possibly constrained discrete variables, are
fundamental to computation. One of the more surprising contributions to
combinatorial optimization over the past decade has come from
statistical physics. Numerous problems, such as graph coloring and
satisfiability, have been analyzed over random instances drawn from an
appropriately parametrized ensemble. (I will define all these
concepts.) What we have learned is that the behavior of optimization
algorithms over such ensembles correlates with an underlying phase
structure: the instances that are computationally hardest to solve are
concentrated near a phase transition. Insights into the connection
between these two phenomena have both transformed our understanding of
the performance of algorithms and inspired new algorithmic methods. I
will discuss results on the graph bisection problem that illustrate the
advantages -- as well as some limitations -- of this approach. For the
case of sparse random graphs, I will give an analytical upper bound on
the optimum cutsize that improves considerably upon previous bounds.
Combined with some recent observations on expander graphs, this bound
leads to two intriguing computational consequences: a) the phase
structure is quite unlike what other combinatorial optimization problems
display, and b) even though the problem is NP-hard, we can likely solve
typical instances near the phase transition in polynomial time.

Where: 
CGU Math South, 710 N. College Ave.

Mathematical Approaches to Studying Molecular Interactions

03/27/2008 - 3:00pm
03/27/2008 - 4:00pm
Speaker: 
Prof. Julie C. Mitchell (U. Wisconsin)
Abstract: 

Proteins, such as enzymes and antibodies, perform many important biological functions. Proteins function by binding to other molecules, and the unique shape and biochemical features of a protein determine its binding partners and hence its function.

Mathematical and statistical approaches to the study of protein interactions can be a useful complement to experiment. We outline how global optimization can be used to predict binding geometries. In addition, we demonstrate how statistical analysis of the geometrical and biophysical features of protein interfaces can be applied toward designing proteins having novel properties, such as enzymes able to selectively kill cancer cells.

Where: 
Burkle 12

Hypermodels in the Bayesian Imaging Framework

02/15/2008 - 2:30pm
02/15/2008 - 3:30pm
Speaker: 
Prof. Daniela Calvetti (Case Western Reserve University)
Abstract: 

We consider the problem of restoring an image from a noisy blurred
copy, with the additional qualitative information that the
image contains sharp discontinuities of unknown size and location.
The flexibility of the Bayesian imaging framework is
particularly convenient in the presence of such qualitative, rather
than quantitative, information. By using a non-stationary Markov
model with the variance of the innovation process also unknown, it
is possible to take advantage of the qualitative prior information,
and Bayesian techniques can be applied to estimate simultaneously
the unknown and the prior variance. Here we present a unified
approach to Bayesian signal and imaging, and show that with rather
standard choices of hyperpriors we obtain some classical
regularization methods as special cases. The application of Bayesian
hyperprior models to imaging applications requires a careful
organization of the computations to overcome the challenges coming
from the high dimensionality. We explain how the computation of
MAP estimates within the proposed Bayesian framework can be made
very efficiency by a judicious use of Krylov iterative methods and
priorconditioners. The Bayesian approach, unlike deterministic
methods which produce a single solution image, provides a
very natural way to assess the reliability of single image estimates
by a Markov Chain Monte Carlo (MCMC) based analysis of the
posterior. Computed examples illustrate the different features and
the computational properties of the Bayesian hypermodel approach to
imaging.

Where: 
Math South, 710 N. College Ave On map http://www.cgu.edu/pages/3233.asp, see location 16

Cost Allocation: Theory and Applications

02/14/2008 - 1:30pm
02/14/2008 - 2:30pm
Speaker: 
Prof. Eric J. Friedman (Cornell University)
Abstract: 

In this talk I will present a personalized survey of some of the mathematics
of cost allocation and its applications. The mathematics will focus on functional
analysis, convex analysis and game theory, both cooperative and non-cooperative.
The applications will include: sharing bandwidth on the Internet, allocating power
on wireless networks, webserving, the manipulation of search engines and even
taxation of multinational corporations.

Where: 
Burkle 12 On map http://www.cgu.edu/pages/3233.asp, see location 10
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