# DE Seminars: Spring 2003

#### Monday February 3

Title |
An A Posteriori Error Estimate for a Nonlinear Elliptic Partial Differential Equation |

Speaker |
Alexandra L. Chaillou UMBC |

**Abstract:**

An a posteriori error estimate is given, as well as a basic introduction to the methods used to obtain a solution to the differential equation. Some numerical results are discussed to demonstrate the accuracy of the estimate in practice.

#### Monday February 24

Title |
Variational Principles, Partial Differential Equations, Anisotropic Diffusion: Can These (Image Processing) Techniques Help to Clean up Gel Electrophoresis Images? |

Speaker |
Jonathan Bell UMBC |

**Abstract:**

My talk is designed to give a few snapshots of some pde-based techniques considered in image processing, with an emphasis on image restoration. The motivation here is the preprocessing of 2D polyacrylamide gel electrophoresis images used to separate and characterize proteins making up biological samples. The applicability of the class of techniques discussed needs to be tested, but the area seems ripe for more modeling, more theory, and better numerical algorithms.

#### Monday March 3

Title |
Idealization of a Chemical Reaction-Diffusion Problem |

Speaker |
Thomas I. Seidman UMBC |

**Abstract:**

This is an updating, since an earlier talk on November 16, 1999 in this Seminar, reporting progress (and still open questions) in the analysis of a singular perturbation problem concerning a compound reaction:

A + B -> C,

A + C -> D

with the first of these extremely fast. Earlier results (joint with L. Kalachev) concerned the problem in steady state. This talk will first comment on those known steady state results, then note some of the interesting questions for the time-dependent problem (including conjectures regarding the idealized solutions), and finally will discuss a few completed pieces of the new analysis.

#### Monday March 31

Title |
Efficient Multiscale Numerical Schemes for Stochastic Bio-Chemical Reactions |

Speaker |
Muruhan Rathinam UMBC |

**Abstract:**

In general many chemically reacting molecular systems are modeled by deterministic and continuous models using ODEs or PDEs. However there are certain situations when the continuous and deterministic models are inadequate. Prime examples are intra-cellular gene transcription mechanisms where the fluctuations of a key molecular species present in very low concentration may have a critical effect on the final state of the system.

On a microscopic level molecular reactions may be modeled by dynamic systems where variables taking discrete values change randomly at random times. In principle such mathematical models can be simulated by keeping track of each “random jump event” of all the variables sequentially as they occur in time. This accurate description however would take prohibitively long computing time for most real problems. In this talk we describe an ongoing project which aims to develop methods, software, and rigorous mathematical theory for simulating these systems in an efficient way by “leaping over” several events at a time.

More broadly these schemes would be applicable to other situations such as the dynamics of market microstructures and traffic models where the detailed models are discrete and stochastic.

#### Monday April 7

Title |
Global Analysis of the Forced van der Pol Equation: The Slow Flow and Canard Solutions |

Speaker |
Kathleen Hoffman UMBC |

**Abstract:**

The forced van der Pol oscillator has been the focus of scientific scrutiny for almost a century, yet its global bifurcation structure is still poorly understood. I will discuss a hybrid system consisting of the dynamics of the trajectories on the slow manifold coupled with “jumps” at the folds in the critical manifold to approximate the fast subsystem. The global bifurcations of the fixed points and periodic points of this hybrid system lead to an understanding of the bifurcations in the periodic orbits (without canards) of the forced van der Pol system. In addition, I will describe a modified hybrid system that includes the canard solutions and briefly describe the additional bifurcations that occur in this extended map.

#### Monday April 14

Title |
Comparisons for Delay Differential Equations |

Speaker |
Thomas I. Seidman UMBC |

**Abstract:**

For delay differential equations, e.g., of the form

dx/t + m x(t) = f(x(t-d)),

we can obtain a comparison theorem much like some more familiar comparison theorems for ordinary differential equations. This can then be used to obtain some asymptotic results. In particular, we provide alternative arguments for some results presented by our visitor from Bangkok, Prof. Yongwimon Lenbury.

#### Monday April 21

Title |
Flow of an Incompressible Viscous Fluid Through a Porous Medium with Periodic Microstructure |

Speaker |
Amber Sallerson UMBC |

**Abstract:**

The purpose of the work is to calculate the permeability of a porous medium from first principles. To do this, the flow of an incompressible viscous fluid through a porous medium with periodic microstructure is analyzed. The fluid flow is modeled as a stationary problem using the Stokes Equations. The microstructure is handled via homogenization. The corresponding unit cell problem can be solved numerically using FEMLAB, a commercial software package for solving partial differential equations with the method of finite elements.

The numerical computations lead to the characterization of the permeability tensor as a function of the porous medium’s void ratio. When the unit cell problem possesses certain symmetries we show that the corresponding permeability tensor is a multiple of identity, therefore the material is isotropic. The numerical calculations confirm the theoretical results. The analysis of permeability of porous media has applications in oil well exploration, gel chromatography, soil mechanics, and biomechanics. This work is part of undergraduate research performed under the direction of Dr. Rouben Rostamian.

#### Monday April 28

Title |
A Parallel Matrix-Free Implementation of the Conjugate Gradient Method for the Poisson Equation in Two and Three Dimensions |

Speaker |
Kevin P. Allen UMBC |

**Abstract:**

The conjugate gradient method is applied to a large, sparse, highly structured linear system of equations obtained from a finite difference discretization of the Poisson equation. This prototype problem is used to analyze the performance of the parallel linear solver on a cluster of workstations. The matrix-free implementation of the matrix-vector product is shown to be optimal with respect to both memory usage and performance. The parallel implementation of the method can give excellent performance on a Beowulf cluster, a group of commodity workstations connected by a dedicated communication network. The optimal number of processors depends on the quality of the interconnect hardware. When only an ethernet interconnect is available, best performance is limited to up to 4 or 5 processors, since the conjugate gradient method necessarily involves several communications per iteration. Using a high performance Myrinet interconnect, excellent speedup is possible for at least up to 32 processors. This justifies the use of the method as the computational kernel for the time-stepping in the numerical solution of a system of reaction-diffusion equations. This work is part of undergraduate research performed under the direction of Dr. Matthias K. Gobbert.

#### Monday May 5

Title |
The Boltzmann Equation: Its Derivation and Application to the Modeling of Microelectronics Manufacturing Techniques |

Speaker |
Steven C. Foster UMBC |

**Abstract:**

Developed in the late-Nineteenth century, the Boltzmann equation is a integro-differential equation. The Boltzmann equation models gas flow, when the mean free path between collisions is on the same or smaller scale than the typical length scale of the domain of interest. Its solution is a probability density function. This presentation will give a brief overview of the derivation of the Boltzmann equation from its physical background, touching on several of the complications as well as the connection to the Navier-Stokes equations. The applications of the Boltzmann equation, however, will be the stress of this talk. The use of the Boltzmann equation to model Chemical Vapor Deposition and Atomic Layer Deposition, both microelectronics manufacturing techniques, and its numerical solution will be discussed. The numerical method for the Boltzmann equation results in a system of linear hyperbolic transport equations. The scalar transport equation is the appropriate prototype problem. A careful performance study has been performed on various computer environments with the results showing adequate to exceptional speedup observed depending on the conditions used in the testing. This work has been conducted under a grant from the Provost’s Undergraduate Research Award (2001-2002) and the supervision of Dr. Matthias K. Gobbert.

#### Wednesday May 7

Title |
A Numerical Study of Nucleation in Stochastic Cahn-Morral Systems |

Speaker |
Jonathan P. Desi UMBC |

**Abstract:**

In metal alloys, one can observe a phenomenon in which impurities nucleate over time. In order to better understand this phenomenon in multi-component alloys, we study a mathematical system introduced by Cahn and Morral. The Cahn-Morral system is a model for several phase separation phenomena in multi-component alloys. In this paper, we study the nucleation of impurities in a stochastic version of this model using numerical simulations. By running many specific simulations of the model, we can obtain much information about the nucleation in multi-component alloys. More specifically, we examine the time it takes for the first impurities to nucleate, the position of these first nuclei, and their concentrations. Through further analysis and more simulations, we can see how changes in composition or certain variables affect the results. Using some statistical analysis of the obtained information, we see how these results relate to the deterministic version of the model. Through this study, we will get a better understanding of this phenomenon and its behavior. This work is part of undergraduate research performed under the direction of Dr. Thomas Wanner.

#### Monday May 12

Title |
Numerical Simulation of Coupled Fluid Flow and Mechanical Deformation Models |

Speaker |
Nicholas Kridler UMBC |

**Abstract:**

In structurally weak geologic formations, production of fluids from oil reservoirs may cause subsidence of the soil. To numerically simulate fluid flow in such regions, we have developed a loosely-coupled one-dimensional computer code that performs both flow calculations and mechanical deformation in staggered-in-time fashion. The elliptic partial differential equation describing deformation and the parabolic pde for flow are solved using the finite element method. Pore pressures from flow act as loads on mechanics, and stress and strain output from mechanics produces dynamically-modified reservoir properties for flow (porosity and permeability). We validate the coupled code against a test problem with known solution — the Terzaghi consolidation problem. In this problem, a thin column of mud is assumed to be impermeable on all sides but the top. A load (book for instance) is placed on top of the column, and fluids drain from the top. The goal of the research is analysis of adaptive time stepping schemes to minimize the number of mechanics steps taken while still producing accurate flow results. The first time-stepping method we implemented involves a comparison of solutions at half and full time steps. Other time-stepping strategies will be compared and analyzed for this problem. This work is part of undergraduate research performed under the direction of Dr. Susan E. Minkoff.