# Math Colloquia: Spring 2010

#### Friday February 05

No talk today

#### Friday February 12

*Cancelled due to weather*

Title |
Multigrid Preconditioning of Linear Systems for Interior Point Methods Applied to a Class of Box-constrained Optimal Control Problems |

Speaker |
Dr. Andrei Draganescu |

Department of Mathematics and Statistics | |

UMBC | |

http://www.math.umbc.edu/~draga/ |

**Abstract:**

In this work we construct and analyze multigrid preconditioners for matrices of the form *D*_{λ} + *K* _{h} K_{h}*, where

*D*

_{λ}is a diagonal matrix respresenting multiplication with a relatively “smooth” function λ and

*K*is a finite element discretization of a compact linear operator

_{h}*K*. These systems arise when applying interior point methods to the distributed optimal control problem

*min*(||

_{u}*K u − f*||

^{2}+ β ||

*u*||

^{2}) with box constraints

*a ≤ u ≤ b*on the control

*u*. The presented preconditioning technique is related to the one developed in earlier work by Draganescu and Dupont for the associated unconstrained problem, and is intended for large-scale problems. The quality of the resulting preconditioners is shown to increase as the resolution

*h*↓

*0*at a rate that is optimal with respect to

*h*if the meshes are uniform, but decreases as the smoothness of λ declines. We test this algorithm first on a Tikhnov-regularized backward parabolic equation with [0,1] constraints and then on a standard elliptic-constrained optimization problem. In both cases it is shown that the number of linear iterations per optimization step, as well as the total number of fine-scale matrix-vector multiplications is decreasing with increasing resolution, thus showing the method to be potentially very efficient for truly large-scale problems.

This is joint work with Cosmin Petra from the Mathematics and Computer Science Division of the Argonne National Laboratory.

#### Friday February 19

Title |
Non-dispersive Wave Arising in the Study of Dispersive Systems and Their Applications |

Speaker |
Dr. Mikhail Kovalyov |

Department of Mathematical and Statistical Sciences | |

University of Alberta, Canada | |

http://www.math.ualberta.ca/Kovalyov_M.html |

**Abstract:**

The KdV/KP equations were derived to describe dispersive waves. Yet they posses non-dispersive solutions, most famous of which are solitons. In the talk we consider another class of non-dipersive solutions. On one hand such solutions appear to explain bore waves and rogue waves. On the other hand they serve as nonlinear analogues of *e ^{i(kx − wt)}* in the sense that their nonlinear superpositions generate a rather large class of solutions of the corresponding equations just like the linear superpositions of

*e*given by the Fourier integral generate solutions of linear PDEs .

^{i(kx − wt)}#### Friday February 26

Title |
FEM and PDEs with Delta Functions as Source Terms |

Speaker |
Dr. Thomas Seidman |

Department of Mathematical and Statistics | |

UMBC | |

http://www.math.umbc.edu/~seidman/ |

**Abstract:**

The standard error estimate for the FEM (Finite Element Method) with piecewise affine elements gives a convergence rate *O(h ^{2})* for the

*L*error in approximation of the Poisson equation

^{2}− Δ

*u = f*when

*f*is in

*L*. For rougher

^{2}*f*(e.g., a delta function or a measure) one expects greater error and exploratory computation suggests an error rate approximately

*O(h*. Theory now justifies this.

^{(2 − dim/2)})[The argument is a combination of Calculus, Geometry, and Functional Analysis. I will attempt to make the exposition understandable by all our graduate students.]

#### Friday March 05

Title |
Match.com or Match.con: Can the NMR and the Signal Processing Matched Filters Find Compatibility? |

Speaker |
Dr. Richard Spencer |

National Institute on Aging | |

National Institutes of Health | |

http://www.grc.nia.nih.gov/branches/lci/nmr/nmr.htm |

**Abstract:**

A type of “matched filter” (MF), used extensively to maximize SNR in NMR spectroscopy, is defined by multiplication of a free-induction decay (FID) signal by a decaying exponential with the same time constant as that of the FID. However, a different entity known also as the matched filter was introduced in the context of pulse detection in the 1940’s and has become widely integrated into signal processing practice. These two types of matched filters appear to be quite distinct. In the NMR case, the “filter” is defined by the characteristics of, and applied to, the time domain signal in order to achieve improved SNR in the spectral domain. In signal processing, the filter is defined by the characteristics of the signal in its spectral domain, and maximizes the SNR in the time (pulse) domain. At face value, then, these filters are distinct, although one suspects an intimate relationship. In particular, since the ideal NMR lineshape, the Lorentizian, is the Fourier transform of a decaying exponential, it is tempting to state that the relationship between the NMR-MF and the signal processor’s MF follows trivially from the convolution theorem. While close, this is not quite the case. Our goal is to reconcile these two distinct versions of the matched filter, demonstrating that the NMR “matched filter” is in fact an example of the matched filter more rigorously defined in the signal processing literature.

#### Friday March 12

No talk today

#### Friday March 19

Spring Break

#### Tuesday March 23

*(note special date)*

Title |
Noncommutative Zero Dimensional Topological Spaces |

Speaker |
Dr. Cornel Pasnicu |

Department of Mathematics | |

University of Puerto Rico, Río Piedras Campus | |

http://math.uprrp.edu/english/personnel/profinfo.php?id=64 |

**Abstract:**

A *C*^{*}-algebra can be thought as a noncommutative topological space or as a collection of infinite matrices of complex numbers endowed with an interesting algebraic and topological structure. The *C*^{*}-algebras have significant applications in different areas of mathematics (geometry, topology, ergodic theory), parts of physics (quantum mechanics and statistical mechanics) and other sciences. Understanding the structure and classification of *C*^{*}-algebras was and continues to be one of the most important researh directions in Operator Algebras (Elliott and Kirchberg, I.C.M. 1994, Rørdam, I.C.M. 2006). In this talk I will present, in a natural context and giving basic definitions and examples, a joint work with Mikael Rørdam (J. Reine Angew. Math. 2007) in which we characterize, in the separable case, for a large and important class of *C*^{*}-algebras that are “infinite” in some specific sense (introduced 10 years ago by Kirchberg and Rørdam) a certain condition of noncommutative zero dimensionality (introduced by Brown and Pedersen) that proved to be very successful in Elliott’s well known Classification Program for *C*^{*}-algebras (I.C.M. 1994). (It is perhaps worth to mention also that many *C*^{*}-algebras of interest happen-sometimes surprisingly-to satisfy this condition). Some interesting consequences of this result that concern the structure of *C*^{*}-algebras will be also discussed. Our theorem strongly generalizes a result of Perera and Rørdam (J. Funct. Anal. 2004) and, in the separable case, a result of Zhang.

#### Friday March 26

Title |
Forecasting Cancer |

Speaker |
Dr. Eric Kostelich |

Deptartment of Mathematics and Statistics | |

Arizona State University | |

http://math.asu.edu/~eric/ |

**Abstract:**

Can cancer be forecast, just as the weather is forecast? This talk explores the relevant mathematics needed to answer this question. Operational meteorological centers update the initial conditions of their forecast models four times a day with noisy measurements of the atmosphere. (Without such updates, the chaotic nature of the atmosphere would yield forecasts that are no more accurate than an almanac’s.) I will review a state-of-the-art data assimilation system, developed with colleagues at the University of Maryland College Park, and describe results with the U.S. Weather Service’s global forecast model. Next, I will consider an application to mathematical biology: replace the weather model with a population model for brain tumor cell growth and atmospheric measurements with magnetic resonance images. I will describe the latest research, done with colleagues at Arizona State University and the Barrow Neurological Institute in Phoenix, to address the feasibility of short-term forecasts of the growth of an individual glioma tumor.

#### Friday April 02

Title |
Numerical Calculations of Leaky Modes in Microstructured Optical Fibers Using Dirichlet-to-Neumann Boundary Conditions |

Speaker |
Dr. Andrew Docherty |

Center for Advanced Studies in Photonics Research (CASPR) | |

UMBC |

**Abstract:**

Microstructured Optical Fibres (MOF), with their flexibility of design and use of diverse materials with special optical properties, are currently of interest for many different applications. Often measures of performance for these fibers are calculated from a knowledge of the effective indices and losses of all or a significant number of the modes – for example the local density of states, bandwidth, numerical aperture, modal cutoff and number of supported modes.

Some different techniques for calculating leaky modes in microstructured optical fiber are outlined, in particular strategies to restrict the numerical calculation to a reduced computational domain. This is achieved by exploiting rotational symmetry and using Dirichlet-to-Neumann boundary conditions.

#### Friday April 09

No talk today

#### Friday April 16

*Join Math/Stat Colloquium*

Title |
Two Groups and One Group Model for Multiple Tests for Microarrays and Other Examples—a Survey and New Results |

Speaker |
Dr. Jayanta Ghosh |

Purdue University |

**Abstract:**

#### Friday April 23

No talk today due to special event: Probability and Statistics Day at UMBC

#### Tuesday April 27

*noon–1pm, UMBC Library, 7th floor*

Title |
Mathematics that Swings: The Math Behind Golf1 ^{st} Chesapeake SIAM Student Chapter Conference |

Speaker |
Dr. Douglas Arnold |

McKnight Presidential Professor of Mathematics, School of Mathematics, University of Minnesota | |

President, Society for Industrial and Applied Mathematics (SIAM) | |

http://www.ima.umn.edu/~arnold/ |

**Abstract:**

Mathematics is everywhere, and the golf course is no exception. Many aspects of the game of golf can be illuminated or improved through mathematical modeling and analysis. We will discuss a few examples, employing mathematics ranging from simple high school algebra to computational techniques at the frontiers of contemporary research.

#### Friday April 30

Title |
Da_{II} → ∞ : A Free Boundary Problem in the Making |

Speaker |
Dr. Thomas Seidman |

Department of Mathematical and Statistics | |

UMBC | |

http://www.math.umbc.edu/~seidman/ |

**Abstract:**

The Damköhler numbers *Da* indicate the balance between chemical reaction rate and material transport (*Da*_{II} when the transport is by diffusion) and, when this is very large, one expects the reactants to be effectively exclusive, never coexisting locally so the region decomposes into subregions where each reactant is available: thus, the reaction may be fast, but only occurs when diffusion brings the reactants together at the interface (free boundary). We are interested here in justifying this intuitive expectation and seeing what happens in the limit as the fast reaction rate goes to infinity.

Here, we consider a model problem involving a compound reaction [*2A+B → **], consisting of an extremely fast reaction [*A+B → C*] (producing the intermediate compound *C*) coupled with a somewhat slower reaction [*A+C → **].

For the steady state problem in one space dimension, it was already known that the solution goes to a well-defined limit as above and we now see how that result can be obtained in the time-dependent context in n spatial dimensions.

#### Friday May 07

Title |
Stationary Distributions for Stochastic Delay Differential Equations with Normal Reflection |

Speaker |
Dr. Michael Kinnally |

Metron |

**Abstract:**

Deterministic dynamic models with delayed feedback and state constraints arise in a variety of applications in science and engineering. There is interest in understanding what effect noise has on the behavior of such models. A stochastic delay differential equation with normal reflection may be considered a noisy analogue of a deterministic system with delayed feedback and positivity constraints. Sufficient conditions for existence and uniqueness of stationary distributions for such equations will be presented, along with applications to a few examples.