DE Seminars: Spring 2007

Monday February 5

Abstract:
Image segmentation, namely identification of boundaries and regions in images, is one of the fundamental problems in image processing. Recently PDE-based methods have found wide use for this problem, specifically within the context of boundary extraction. One of the pioneering models of this approach is the Geodesic Active Contour Model introduced by Caselles, Kimmel and Sapiro. This is essentially a weighted length formula (or energy) for curves. The weight is defined such that curves on boundaries of objects in a given image correspond to local minima of this energy. In this approach, the goal is to start with an initial curve and compute deformations (velocities) to obtain an energy-minimizing sequence of curves. In this talk I will describe a method based on finite elements to achieve this. An important feature of the method is the flexibility to work with different velocity spaces, possibly smoother and faster than traditional choices. Space adaptivity and topological changes, such as merging and splitting of curves, are incorporated as well. The mathematics naturally extends to surfaces in 3d. I will demonstrate the method with several examples.

Monday February 19

Abstract:
Compatible discretization methods for solving partial differential equations are often used in electromagnetics to preserve the underlying physical properties expressed in the equations. The mimetic method of J. Hyman and M. Shashkov and the covolume method of R. Nicolaides are two compatible discretization methods that rely on a discrete vector calculus structure. These two methods can be placed in a common framework through the use of discrete differential complexes. In this talk we describe this framework in two and three dimensions, how it captures fundamental relationships among first- order differential operators, and its connection to the finite element methods of Raviart-Thomas and Nedelec, and demonstrate how convergence results for div-curl systems can be extended to a larger class of algorithms.

Monday March 5

Abstract:
In this talk I will first give an overview of the current research in my laboratory, the Dynamic Systems and Vibrations Laboratory in the Department of Mechanical Engineering at UMBC. Our research combines analytical development, numerical simulation, experimental validation, and industrial applications. Some theoretical problems studied include analysis of time-varying and infinite-dimensional systems, nonlinear analysis, stochastic analysis, inverse modeling, and joint modeling. Experiments are designed and conducted using the state-of-the-art equipment to validate the theoretical predictions. The research can be applied to the design of elevators, belt and tape drives, and power transmission lines. It also has applications in nondestructive testing and modal testing. I will then focus on: 1) a new dynamic stability theory for translating media with variable length and/or speed; 2) a robust iterative algorithm for structural damage detection using a minimum number of vibration measurements, along with new physics-based methods to model structures with L-shaped beams and bolted joints; and 3) a novel stochastic model for the random impact test method in modal testing. Experimental results on a novel scaled elevator, damage detection, and joint modeling will be demonstrated.

Monday March 26

Abstract:
This talk will introduce infectious disease modeling within the context of ordinary differential equations. The first part will introduce .SIR. models, with a focus on the biological interpretation of terms and parameters appearing in the models. The second part will illustrate these methods using Rift Valley fever (RVF) — a mosquito-borne viral disease of humans and livestock in Africa & the Middle East, and an important biological threat to the Western Hemisphere. Results of the model will be described, including application to applied problems in public health.

Monday April 2

Abstract:
In this talk we discuss the solvability of the Poisson problem in weighted Sobolev spaces, in two-dimensional domains with conical points. We follow the method of V. A. Kondrat’ev and derive regularity theorems in the context of these weighted Sobolev spaces and obtain expansions of the solution in the neighborhood of the conical points.

Monday April 9

Abstract:
Data assimilation is the methodology to assimilate observational data with numerical models for better estimation of the true physical states. This project focuses on application of an error covariance, generated by an ensemble method, to assimilate surface geomagnetic data to numerical geodynamo model, aiming at better understanding the dynamical processes in the Earths core and at predicting geomagnetic secular variation on decadal time scales and longer. This project can be divided into two parts: (i) understanding the data quality and the numerical forecast with a simplified MHD system (to get the first-hand knowledge necessary for the full geodynamo system), and (ii) assimilating synthetic geomagnetic data to the full geodynamo model through a series of Observing Simulation System Experiments (OSSEs). In particular, through the project, we wish to understand how the full dynamo state (from numerical model) are affected (or corrected) by limited surface observations, and whether a frozen (in time) covariance could be sufficient to bring analysis closer to truth. The latter is in particular important for geomagnetic data assimilation. Should it be sufficient, then the computation needs for geomagnetic data assimilation can be reduced by one order of magnitude. Research results suggest that (i) sparse observation could produce a significant constraint on the numerical model to make the forecast closer to the true physical states; (ii) observed physical variables correlate strongly with other unobservable fields in the dynamo process, such as the poloidal magnetic field (observable) and the toroidal magnetic field (not observable), and appropriate implementation of the cross-correlation could improve the assimilation system; (iii) dynamo solutions can converge to the surface observations in a very short time period (compared to the magnetic free-decay time); however, the convergence of the dynamo solutions in the deep interior occurs in much longer time periods.

Monday May 14

Abstract:
I will describe some of our research at the Naval Academy toward understanding the hydrodynamics of the Chesapeake Bay. I will begin with presenting the underlying PDEs of motion and describe two attempts to solve the initial-boundary problem numerically. I will also describe dynamical systems tools that have recently been developed that are applied to the numerically computed velocity fields in order to characterize coherent structures in the flow.