APPLIED MATH SEMINARS
ABSTRACTS

T. Hangartner : Correction Procedures in Quantitative Computed Tomography

Computed tomography is a radiologic method that allows the visualization of thin cross-sections through the body. Whereas most images are viewed by a radiologist and assessed in a qualitative way, there are applications that endeavor to extract quantitative information about the tissue. In such applications, it is important that the generated images are free of errors, and special care needs to be taken to understand the discrepancies between the assumptions in measurement and reconstruction process and the physical effects in the actual instrument. Once the effects are understood, it may be possible to develop appropriate correction procedures. The following effects will be discussed and suggestions for corrections will be outlined: photon background, detector dead time, beam hardening, scatter and blurring. Some effects can be modeled theoretically, and analytical corrections can be devised. Other effects can only be simulated approximately and require iterative corrections. In the whole process of selecting the most appropriate measures to reduce the quantitative errors, one needs to balance the need for accuracy with the impact on the patient with respect to radiation dose and measurement time.

D. Watts : Seismic data processing using wavelet transforms.

Seismic reflection data are often contaminated by natural and source generated noise. Indeed, many parts of the world remain impossible to image using seismic methods because the noise dominates seismic records. Fourier methods are limited by the non stationary nature of seismic data and the noise contained on these records. The common trick in signal processing employed to remove noise is to transform the data to a domain where the data separates from the noise, replace the offending part of the transform domain with zeros and apply the inverse transformation. Wavelet transforms and the related wavelet packet transforms are useful for removing certain types of noise on seismic records. They are most useful for the noise that is confined to certain time distance windows on shot records. The most useful wavelets for windowed filters appear to be the Battle-Lamarie family of wavelets.

T. Skinner : Classical Rotations and Quantum Spin Dynamics in Nuclear Magnetic Resonance (NMR)

Quantum mechanics (QM) provides an unerringly accurate description of NMR phenomena. Given sufficient computational resources, the results of any NMR experiment can be calculated with precision. Although the theory needs no supporting visual model for its application, such models can allow one to intuit correct results and predictions for new experiments "on-the-fly," a useful capability that is often precluded by a pure resort to the abstract machinery of QM. An overview of basic NMR is first provided in the context of the classical Bloch equation, which describes the torque on a magnetic moment in an external magnetic field, resulting in a constant precession of the magnetic moment vector about an axis defined by the field vector. This model for NMR of systems composed of a single nuclear species (such as protons or carbon-13, for example) is easily pictured and provides an accurate solution for the time evolution of the system. However, this simple model cannot accomodate more complicated situations such as the indirect (scalar) coupling between two spin-1/2 nuclei. The indirect coupling plays an important role in modern high resolution NMR, so a brief discussion illustrating its importance is provided next. Of special relevance are cases where radio frequency (RF) irradiation is applied to one of the spin species. The QM equation of motion necessary for a rigorous analysis of general NMR systems is then considered. These equations are solved to obtain the time dependence of the density operator for coupled systems during RF irradiation of one of the nuclear species. Previous solutions for the evolution of the density operator considered either evolution in the absence of RF irradiation or the case where the RF is sufficiently strong that indirect coupling effects could be ignored. The new solutions have yielded several novel applications for high resolution NMR which will be discussed. At first glance, the solutions appear slightly complicated and representative of non-classical phenomena. However, an alternative analysis leads full-circle to a simple "Bloch equation" for coupled systems in NMR and to a rigorous derivation of the original Bloch equation from the QM equation of motion. The utility of a resulting vector picture for nuclear spin evolution of coupled systems is illustrated.

Dr. V.B. Venkayya : Multidisciplinary Technologies (MDT): New Research Center at AFRL

Multidisciplinary integration for airframe structures design is an active research area at the Air Force Research Laboratory (AFRL). Aerodynamics, Structures and Controls are some of the disciplines considered for integration in the framework of mathematical optimization to achieve the performance goals of Air and Space Vehicles. The process is called Multidisciplinary, because the objectives and constraints are derived from all the participating disciplines and the search for the optimum is formalized through the use of gradient and/or non-gradient algorithms. Mathematical modeling and simulation of the physical systems (vehicles) response plays a key role in this research effort. The seminar will highlight some of the issues (challenges) in this process.

G. Kozlowski : Magnetocaloric Effect in Magnetic Materials

Large cooling by adiabatic magnetization (magnetocaloric effect) of antiferromagnetic Ytterbium and Gadolinium garnets has been found by Clark and Callen. They observed exceptionally large cooling from 18 K to 8 K by increasing the magnetic field to 11 Tesla. This talk will review my earlier theoretical work on this effect in antiferromagnets and ferromagnetic superconductors. The dependence of the effect on the strength of the exchange and anisotropy interactions in antiferromagnetic materials including interaction with phonons will be discussed by using the low temperature approach (spin wave theory). A more phenomenological approach will be used for ferromagnetic superconductors with emphasis on the phase diagram which describes magnetic changes in the system due to temperature and external magnetic field.

T. Svobodny : Magnetocaloric Effect with Magnetoelastic Interaction

The Magnetocaloric Effect with Magnetoelastic Interaction. This talk will be a bridge between those of Skinner and Kozlowski. I will introduce the use of statistical mechanics in the interaction of a magnetic system with a material lattice. After showing how the entropy is calculated from the energy of small excitations, I will point out an analogy to the vortex lattice of a type-II superconductor. This talk is based on the annotated version of a paper in Stat. Solidi.

C. Huang : Regularity of the Lake Equations

We shall study the vorticity formulation for the lake equations that models the circulation of invscid fluid in a very large shallow basin with a varying bottom. Under the assume that initially the vortex region is small and isolated, we show that the problem can be formulated as a integral system. Global existence and uniqueness of classical solutions to the integral system are established. Consequently, the vortex region remains as smooth as its initial state.

R. Collins : Computational simulation of stable atmospheric boundary layers

Stable atmospheric boundary layers (ABL's) occur mainly at night, when the ground cools. Flows are typically laminar, with intermittent episodes of strong turbulent mixing. The resulting turbulence is not of the statistically steady-state type on which existing theories are based. The Army Research Lab (ARL) wishes to predict transport and diffusion of chemical and biological agents in such stable boundary layers. The underlying turbulent transport mechanisms responsible for the turbulent bursts are not yet clearly understood. Current LES (large eddy simulation) does not capture these flow characteristics well. We propose to render such predictions more reliable by combining experimental data with novel computational modeling initiatives. A succinct focussed review of this problem will be presented.

K. Tomko : Domain decomposition for applications with structured and unstructured grids

Parallelization of scientific simulations typically requires a decomposition of the simulation domain and a mapping of the subdomains to processors. I will discuss techniques and issues in partitioning a 3-D Navier-Stokes solver which uses structured overset grids. I will also cover graph partitioning techniques used with applications which utilize unstructured grids such as Finite Element based applications and discuss some of the issues in adapting graph partitioning methods to networks of workstations.

D. Roberston : High-Performance Computing at OSC: An Overview

I present an overview of the high-performance computing resources available to Ohio academic researchers through the Ohio Supercomputer Center. As a state-supported institution, OSC's resources are subsidized through the Ohio Board of Regents, and are available to all Ohio academic researchers and their collaborators. I describe the hardware and software environments currently available, providing a brief introduction to vectorization and a survey of the various paradigms for parallel computing. I also describe the procedures for getting access to OSC supercomputers, and other available resources such as training, consultation and advanced scientific visualization.

T. Knox : Too much of a good thing: In pursuit of human tolerance criteria

Biodynamics is a discipline which focuses on the response of the human to various forces. Knowledge developed during the study of biodymanics is applied to improving safety of vehicles (bicycles, cars, trucks, watercraft, aircraft and spacecraft), helmets and escape systems. This talk will introduce the work being conducted by the Biodynamics and Acceleration Branch of Air Force Research Lab at Wright-Patterson AFB with emphasis on the development of human tolerance to abrupt acceleration. Both experiments (rocket sleds, ejection seat tests, laboratory accelerators, cadavers, manikins, human volunteers, racecars, contact sports) and modeling (ATB model, DRI, MDRC, Finite Element) will be discussed.

B. Foy : Diffusion Coefficients of Proteins in Cartilage

As a relatively avascular tissue, articular cartilage and chondrocytes must receive nutrients and macromolecules by diffusive transport. Thus, quantifying the transport of molecules in healthy and damaged, osteoarthritic cartilage has relevance both for the progression of osteoarthritis, and for potential drug treatments. The diffusion coefficient and relaxivity of paramagnetic compounds in gels and cartilage is measured using a magnetic resonance 1-dimensional imaging technique. The paramagnetic compound to be studied can be either an element such as copper or gadolinium, or the paramagnetic element can be bound to a larger molecule such as a protein.The technique relies on the effect of the paramagnetic on the T1 relaxation properties of the surrounding water. Thus as the concentration of the paramagnetic rises, due to diffusion of the paramagnetic into a material, the NMR signal experiences a faster relaxation rate. Acquisition of the NMR signal is performed using an inversion recovery pulse sequence. The technique is an extension of a previously published NMR technique, with the additional capability of measuring the relaxivity of the paramagnetic in the gel or cartilage matrix. The resulting diffusion coefficient data is most easily interpreted by calculating the ratio of the effective diffusion coefficient of a compound in cartilage to its diffusion coefficient in water. The results of macromolecular diffusion in cartilage samples indicate that larger molecules exhibit a smaller ratio than small molecules. This ratio provides clues on pore sizes and charge interactions in the cartilage. Upon treating the cartilage with an enzyme that degrades the cartilage in a manner similar to osteoarthritis, the diffusion coefficients are markedly elevated.

J. Haus : Photonic band structures: analysis and computations

A new field of research has emerged with materials patterned into periodic dielectric structures called photonic band structures. They provide new conceptual solutions to bottlenecks in optical and electronics communications. In this talk I review some of the mathematical techniques and tools we use to understand the basic electromagnetic properties of these designer materials.

C. Pettit : Uncertainty-Based Modeling of Nonlinear Systems for Multidisciplinary Integration

Optimum design problems in aerospace engineering typically are subject to conflicting design constraints from multiple disciplines. A common example is the minimum weight design of an aircraft structure that must satisfy several aeroelastic response criteria. Traditional multidisciplinary design tools couple linear finite element and aerodynamics models because of their computational efficiency, which is of paramount concern in an iterative optimization process. However, future Air Force vehicles must extend current technological boundaries if the U.S. is to maintain air superiority. This mandate requires more efficient and dependable operation of aircraft in nonlinear response regimes, for which our existing design tools are inadequate. These problems are exacerbated by omnipresent, but poorly understood, levels of uncertainty in system parameters, boundary conditions, and modeling fidelity. Current practice compensates for these inadequacies through safety margins on critical response quantities (stresses, modal frequencies, etc.), which result in structural designs with unknown levels of safety and robustness.

The primary objective of our work is to develop strategies for analyzing multidisciplinary nonlinear systems subject to uncertain initial conditions, boundary conditions, and system parameters, so that the use of high-fidelity mathematical models can be made tractable for optimizing complex dynamic systems. Use of these tools will promote better understanding of the response bounds for these complex systems, as well as their sensitivity variations in uncertain parameters. Applications to be discussed the include quantification of atmospheric turbulence for input to computational fluid dynamics (CFD) models, and the CFD-based calculation of control surface deflections needed to trim an aircraft in a turbulent atmosphere.

Dalta V. Gaitonde : High-Order Methods for Practical Multi-disciplinary Simulations

This presentation will discuss recent successes in developing a set of versatile schemes to solve multi-disciplinary problems on complex geometries. Pade-type fourth- and sixth-order methods have been incorporated in implicit and explicit time-marching strategies together with a newly developed high-order filtering technique. This key component overcomes previous limitations associated with numerical instability arising from non-linearity, boundary condition implementation and mesh non-uniformity. Complementary extensions to treat moving meshes and multi-domain strategies have yielded a powerful common platform to solve problems in fluid dynamics, electromagnetics, magnetogasdynamics and fluid-structure interactions. Several results obtained in the Computational Sciences Branch will be described and future direction will be outlined.

John F. Maguire : Computer Modeling in Materials Design and Discovery: Successes and Challenges

This talk will review the evolution of computer simulation methods, primarily the Monte Carlo and molecular dynamics approaches, to solution of the N-body problem in statistical mechanics. The fundamental difference between "soft" and "hard" potential functions will be highlighted as well as the need to establish more rigorous approaches to fundamental problems in the space-filling geometry of convex objects. The extension of the hard-sphere and hard-rod work to objects of arbitrary geometry (illustrated by simulation of 64000 hard space- filling tetrahedra) has been achieved through an interesting application of artificial neural nets. These new techniques enable essentially exact model simulations of complex systems such as new advanced aerospace materials, as well as everyday objects such as heaps of sand and avalanche. Applications in other fields such as collision detection in aerospace, automotive, and battlefield simulation will be discussed briefly.

Qingbo Huang : Fully nonlinear elliptic equations in generalized John-Nirenberg spaces

Fully nonlinear elliptic equations $F(D^2u, x)=f(x)$ arises in many areas of mathematics and sciences such as differential geometry, stochastic control, differential games, finance, engineering, etc. In the past two decades, the investigation of these equations has undergone a rapid development. When an equation is concave or convex and given data are smooth, the theory of classical solutions has been well developed. However, to deal with general situation, the notion of viscosity solutions was introduced. It is well known in Caffarelli's work that $W^{2,p}$ estimates and $C^{2,\alpha}$ estimates hold for these equations under certain assumptions. One interesting question is that, can we establish a regularity theory for the intermediate case between $L^p$ and $C^{2,\alpha}$? In this talk, we will discuss recent development about estimates in BMO and generalized John-Nirenebrg space BMO${}_{\psi}$ for hessians of solutions to fully nonlinear uniformly elliptic euqations. This result, together with $W^{2,p}$ and $C^{2,\alpha}$ estimates, gives the regularity theory in the full scope from $L^p$ and $C^{2,\alpha}$.