USNA Applied Mathematics Colloquium

Thursday, March 20, 2008, 12:00 noon Dr. Lisa Oberbroeckling (Loyola College)

An hp finite element method for singularly perturbed systems of reaction-diffusion equations

abstract: We consider the approximation of a coupled system of two singularly perturbed linear reaction-diffusion equations using the finite element method. The solution to such problems contains boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. We present results on a high order $hp$ finite element scheme which includes elements of size $O(p \varepsilon)$ and $O(p \mu)$ near the boundary, where $\varepsilon$ and $\mu$ are the singular perturbation parameters and $p$ is the degree of the approximating polynomials. Under the assumption of analytic input data, the method converges at an exponential rate, as $p \rightarrow \infty$, independently of $\varepsilon$ and $\mu$. Numerical results which validate the analysis and compare the proposed method with other schemes found in the literature will also be presented.

Thursday, March 27, 2008, 12:00 noon Dr. Radu Balan (Univ. of Maryland)

Sparse Component Analysis: use of statistical methods and sparse signal representations in convolutive blind source separation problems

abstract: Sparse Component Analysis represents an overlap of two problems (and methods) of Statistics/Computer Science/Electrical Engineering/Applied Mathematics: Independent Component Analysis (ICA), and Sparse Representations. Originally, the ICA problem is looking for decomposing a random d-vector into a linear compoisition of exactly d independent random variables: x = A.s , where A is dxd unknown mixing matrix, and s is the d-vector of independent components. The Blind Source Separation (BSS) problem is very much similar to ICA, except that A may be a matrix of (convolutive) operators. In practice, people applied these solutions to different type of signals. In particular audio (speech) signals gave rise to what is also known as "the cocktail party problem". Interesting algorithms were also obtained on images, bio-medical signals (e.g. EEG, ERP, fMRI). Independent of this, the Sparse Representation problem tries to decompose a vector x into a linear combination of (possibly redundant) frame vectors using a smallest number of coefficients. My talk uses sparse representation hypotheses in order to solve a convolutive BSS, including estimating the number of source signals.

Thursday, April 3, 2008, 12:00 noon Dr. Andrei Draganescu (UMBC)

Multigrid methods for large-scale inverse problems

abstract: Scenario: An air contamination event takes place in a heavily populated area. A chemical agent is being diffused in the air and moved by the wind. Sensors monitoring air quality detect increased concentrations of the pollutant. At what location was the pollutant released in the air? What areas will be affected over the next few minutes, hours, days, and to what degree?

Fast answers to these questions are critical for hazard containment and assessment, and for evacuation strategies. An efficient response to the above scenario requires the backward solution of a time-dependent advection-reaction-diffusion equation, a problem that is ill-posed. In this talk I will present a method for efficiently solving the regularized inverse problem of identifying initial conditions given various measurements in space and time for an equation of parabolic type. The method is a new embodiment of the multigrid paradigm, which consists in using several discretizations of an equation in order to speed up the solution process. I will discuss mathematical results both for linear and nonlinear models, and I will present supporting numerical results. The method is applied to the above scenario, showing that the associated high-resolution inverse problem can be solved in a reasonable time-frame.

Thursday, April 10, 2008, 12:00 noon on the first floor Dr. Padmanabhan Seshaiyer (GMU)

Multilevel computational methodologies for solving fluid-structure interactions with applications to biological and bio-inspired systems

abstract: In the last few years, there has been a great demand to develop novel computational strategies to better understand multi-physics problems such as fluid-structure interaction that arise naturally in various engineering and biological applications. Efficient solution methodologies to the associated coupled system provides predictive capability in studying complex nonlinear interactions that arise in several biological applications (such as an aneurysm interacting with blood flow) and several bio-inspired systems (such as Micro Air Vehicles). This talk will present an overview of a variety of analytical, computational and experimental techniques used to model, analyze and solve such coupled problems efficiently. Numerical results will also be presented to demonstrate the robustness of the proposed techniques.

Thursday, April 17, 2008, 12:00 noon Dr. John T. Spyropoulos (NAVAIR)

Exhaust jet noise reduction for tactical naval aircraft

abstract: Structural airframe/nozzle integrity issues due to ever-increasing acoustic loads have not yet resulted in procurement requirements to subdue the exhaust jet noise of tactical naval aircraft propulsion systems. Issues with noise physiological limits for carrier deck launch/recovery support personnel cannot be addressed by current/future protection gear resulting in hearing loss health claims at escalating costs. Issues with far field noise pollution for NAS surrounding communities can only be addressed short-term by relocating the air bases to more remote locations at great political/monetary costs. Combined, these issues are imposing de-facto derived procurement requirements on engine noise that are being ignored driving up system life cycle cost. In addition, survivability concerns have imposed future program requirements (JSF) that address the associated issue of thermal signature. The talk provides an overview of the many aspects of the exhaust jet noise problem from the propulsion community perspective with proposed roadmap for fundamental research and noise reduction technologies for the short, medium and long term.