# DE Seminars: Fall 2008

#### Monday September 22

Title |
Interface conditions for a singular reaction/diffusion system |

Speaker |
Thomas Seidman Department of Mathematics and Statistics UMBC |

**Abstract:**

If one chemical reaction $A+B \to C$ is very much faster than any others in a diffusion/reaction system, then the reaction will be confined to a narrow reaction zone, with $A,B$ fed to this zone by diffusion. In the limit (reaction rate taken as infinite) the system becomes singular as the reaction zone narrows to a sharp interface. We consider a model problem of this type, emphasizing the determination of the conditions governing the evolution of the interface as a free boundary. remark: I last spoke at UMBC about this analysis (then restricted to the steady state problem) in 2003 so this may be viewed as an update.

#### Monday October 13

Title |
Diffusion in Media with Random Microstructure |

Speaker |
Alen Agheksanterian Department of Mathematics and Statistics UMBC |

**Abstract:**

We consider modeling of diffusion in a medium with heterogeneous microstructure. Such a medium can often be described by its effective behavior; this entails replacing the real heterogeneous medium with a homogeneous medium . the effective medium . whose diffusive properties closely approximate those of the real medium. A process of averaging or homogenization takes place so that the complicated microstructure of the medium is replaced by an asymptotically equivalent homogeneous structure. The goal of this talk is to give a brief presentation of some basic results in homogenization theory applied to a second order elliptic problem of form -div(A * grad u) = f where the diffusion matrix A is a random function.

#### Monday October 20

Title |
Trader behavior and its effect on asset price dynamics |

Speaker |
Muruhan Rathinam Department of Mathematics and Statistics UMBC |

**Abstract:**

In the modeling of stock markets it has been common practice to model the stock price process from a phenomenological point of view and traders are assumed not to influence this process. In this talk we present a first principles framework that models the interaction between trader behavior and the stock price dynamics. Our model starts with a continuous time discrete event description and under suitable scaling a coupled system of stochastic differential equations are derived. In addition to extraneous traders, we consider traders with three different types of trading strategies: value, momentum, and hedge traders. We show via analysis and numerical simulations that this simple model can explain some of the stylized features of the stock market such as stochastic volatility, volatility clustering, as well as the stock pinning phenomenon.

This is joint work with Dr James A. Primbs.

#### Monday October 27

Title |
Efficient numerical methods for stiff systems of stochastic differential equations |

Speaker |
Ioana Cipcigan Department of Mathematics and Statistics UMBC |

**Abstract:**

In deterministic as well as stochastic models, stiff systems, i.e., systems with vastly different time scales where the fast scales are stable, are very common. It is well known that the implicit Euler method is well suited for stiff deterministic equations (modeled by ODEs) while the explicit Euler is not. In particular, once the fast transients are over, the implicit Euler allows for the choice of time steps comparable to the slowest time scales of the system. In stochastic systems (modeled by SDEs) the picture is more complex. While the implicit Euler has better stability properties over the explicit Euler, it underestimates the stationary variance. In general, one may not expect any method to work successfully by taking time steps of the order of the slowest time scale. We explore the idea of interlacing large implicit Euler steps with a sequence of small explicit Euler steps. In particular, we study the uniform convergence with respect to time scale separation parameter for a linear test system and demonstrate that such interlacing could effectively deal with stiffness.

This is joint work with Dr. Muruhan Rathinam.

#### Monday November 10

Title |
A receiver model for optical communications systems with polarization effects |

Speaker |
John Zweck Department of Mathematics and Statistics UMBC |

**Abstract:**

I will derive an analytical formula describing the performance of optical fiber communications systems in which polarization effects play a significant role. I will show excellent agreement between the formula, Monte Carlo simulations, and laboratory experiments. The theory will be used to assess the variation of the bit-error ratio due to random fluctuations in the system when the signal is affected by polarization-mode dispersion and the noise is partially polarized due to polarization-dependent loss.

This work is joint with CSEE graduate student Hua Jiao, with assistance from Professors Gary Carter, Curtis Menyuk, and Li Yan (CSEE, UMBC).

#### Monday November 17

Title |
Multilevel methods for inverse problems with bound-constraints |

Speaker |
Andrei Draganescu Department of Mathematics and Statistics UMBC |

**Abstract:**

Recently developed multilevel (ML) methods have been shown to solve certain large-scale inverse problems at a cost that is comparable (in order of magnitude) to the cost of solving the associated direct problem. More precisely, the following qualitative behavior has been established: for a class of problems that includes the unconstrained (regularized) backwards heat-equation, specialized ML algorithms show the relative cost of solving the inverse problem versus that of the direct problem to be decreasing with increasing problem resolution.

The addition of explicit bound-constraints in the formulation of the inverse problem can be critical for rendering physically meaningful solutions, and it is also shown to significantly improve the quality of inversion. In this talk we address the question of whether the qualitative behavior of ML methods established for unconstrained inverse problems can be replicated for inverse problems with bound constraints.

#### Monday November 24

Title |
Dynamics of Langmuir films: Theory and experiment |

Speaker |
James Alexander Department of Mathematics Case Western University |

**Abstract:**

A Langmuir film is a molecularly thin film on a surface. Here we consider the behavior of such films on a flat water surface. (Note: “we” includes a physicist, Elizabeth Mann, a chemical engineer, Jay Mann, another mathematician, Andrew Bernoff, as well as several students.) The equilibrium state of a finite domain (= “blob”) of such a film is a circular disk. We consider the dynamical behavior when such a domain is disturbed. The tension at the boundary of the domain is the restoring force (so-called line tension, analogous to surface tension of a fluid), which competes against the viscosity of the underlying fluid. A prominent problem is to measure the line tension. The system is modeled as two coupled Navier-Stokes systems, which, because the Reynolds numbers are small, reduces to two coupled Euler flows. Solutions are found which permit comparison with experiment. An advance we make is to model the system globally, not just asymptotically at the end of relaxation to equilibrium. This requires numerical computations. A priori, since pressure mediates the dynamics, a 2-d time dependent numerical integration seems to be required. However, the integrals can be reduced to 1-d boundary integrals, making computation effective. In this lecture, we describe the project, from conceptual beginning through to the quantitative dovetailing of theory with experiment.

Some published results:

- Determination of interphase line tension in Langmuir films (with Jacob R. Wintersmith, Lu Zou, Andrew J. Bernoff, J. Adin Mann Jr., Edgar E. Kooijman, Elizabeth K. Mann), Phys. Rev. E, Phys. Rev. E, 75(6), June, 2007.
- Hole dynamics of polymer Langmuir films (with Andrew Bernoff, Elizabeth K. Mann, J. Adin Mann, Jr., Jacob Wintersmith, Lu Zou), J. Fluid Mech. 571(1) (January 25, 2007), 191-219 (with online supplementary material).
- Domain relaxation in Langmuir layers (with Andrew Bernoff, Elizabeth K. Mann, J. Adin Mann, Jr., Lu Zou), Phys. Fluids 18(6) (June 2006) (with online supplementary material).

#### Monday December 1

Title |
Theoretical Analysis of a Quartz-Enhanced Photoacoustic Spectroscopy Sensor |

Speaker |
Noemi Zakarias Department of Mathematics and Statistics UMBC |

**Abstract:**

Quartz-enhanced photoacoustic spectroscopy (QEPAS) sensors are based on a recent approach to photoacoustic detection which employs a quartz tuning fork as an acoustic transducer. These sensors enable detection of trace gases for air quality monitoring, industrial process control, and medical diagnostics. To detect a trace gas, modulated laser radiation is directed between the tines of a tuning fork. The optical energy absorbed by the gas results in a periodic thermal expansion which gives rise to a weak acoustic pressure wave. This pressure wave excites a resonant vibration of the tuning fork thereby generating an electrical signal via the piezoelectric effect.

In the first part of my talk, I will describe a theoretical model for the most basic configuration of a QEPAS sensor. By deriving analytical solutions for the partial differential equations in the model, we obtain a formula for the piezoelectric current. We use the model to study the sensitivity of the sensor relative to the system parameters. Simulation results which closely match experimental data will be presented. In the second part of my talk, we will look at a more complicated configuration of the QEPAS sensor, in which the tuning fork is augmented with a cylindrical microresonator to enhance the sensitivity of the sensor. Acoustic problems in complicated unbounded domains are often modeled by a boundary integral equation (BIE). The essential feature of such method is that it reduces the problem to a bounded domain of lower dimension. The goal of this part of the talk is to give a brief presentation of the boundary element method and describe how it can be used to approximate the solution to time harmonic acoustic problems.

#### Monday December 8

Title |
Comparison of Parallel Performance between MVAPICH2 and OpenMPI Applied to a Hyperbolic Test Problem |

Speaker |
Michael Reid Department of Mathematics and Statistics UMBC |

**Abstract:**

During the manufacture of integrated circuits, the process of atomic layer deposition (ALD) is used to deposit a uniform seed layer of solid material atop the surface of a silicon wafer. The process can be modeled on the molecular level by a system of transient, linear integro-partial differential Boltzmann equations, coupled with a non-linear surface reaction model, together called the kinetic transport and reaction model (KTRM). Each Boltzmann equation can be approximated by discretizing the velocity space, which yields a system of transient hyperbolic conservation laws that only involve the position vector and time as independent variables. The system can then be solved with DG, a computer implementation of the discontinuous Galerkin method. Due to the large size of the systems being solved and large number of time steps required, it is necessary to use parallel computing to obtain a solution in a reasonable amount of time. We analyze the performance of the DG code on multiple mesh resolutions by measuring its speedup and efficiency on UMBC~Rs new distributed-memory cluster, hpc (www.umbc.edu/hpcf). We also compare the performance of DG when it is compiled using the MVAPICH2 and OpenMPI implementations of MPI, the most prevalent parallel communication library today.

This work is part of undergraduate research conducted under the direction of Dr. Matthias K. Gobbert.