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

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

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:

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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

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)

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.

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

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

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