Table of Contents
Research in Computational
|Finite Element Computations||Dynamical Systems|
|Fluid Mechanics||Numerical Methods for ODEs|
|Nonlinear Waves||Boundary Element Computations|
Professor Aceves' research focuses on the mathematical modeling of phenomena in nonlinear optics. By using techniques from perturbation theory and asymptotics together with numerical simulations, this research studies the pulse dynamics in nonlinear optical media. Most recent work includes light localization in optical fiber arrays, trapping of light in waveguide gratings and ultraviolet light filament formation in the atmosphere.
Besides working with his current PhD students, other collaborations include scientists
at the Air Force Research Laboratory, the Department of Physics and Astronomy at the
University of New Mexico and with the Electromagnetism and Photonics group at the
Universita di Brescia in Italy. Recent work has appeared in Physica D, Studies in
Applied Mathematics, Physical Review A, and Optics Communications. For the past five years,
he has been an affiliate of the Mathematical Modeling and Analysis group (T7) of the
Theoretical Division at the Los Alamos National Laboratory.
Professor Ajaev's research involves applications of asymptotic and perturbation as well as numerical methods for partial differential equations to various problems in fluid mechanics and crystal growth. Of particular interest are simulations of moving interfaces in systems with phase transitions by means of finite-difference and boundary-integral methods.
He collaborates with scientists at the University of California, Santa Barbara
and the Technical University of Darmstadt, Germany. His published work has appeared
in Annual Review of Fluid Mechanics, Journal of Fluid Mechanics, Physics of Fluids,
Physical Review E, Journal of Computational Physics, Numerical Heat Transfer,
Proceedings of the Royal Society A, Journal of Crystal Growth,
and Journal of Colloid and
Professor Barreiro's research is in mathematical modeling, analysis and simulation of neural networks. She is particularly interested in how network architecture and dynamics combine to produce correlated activity (or synchrony) in neural "microcircuits", and in the resulting consequences for computation. Such microcircuits form the building blocks of biological neural circuits, and thus the operation of the nervous system. The ultimate goal of this work is to give insight into the fundamental architecture of how the brain works, and it has the potential to address models of diseases that manifest as excessive synchrony, such as Parkinson's disease. Some of the tools she uses for this work are dynamical systems, probability and stochastic processes, information theory, perturbation methods and numerical simulations. She also works on models of neural integrators, which allow the brain to store evidence and memories, on numerical methods for neural population dynamics, and on analytical and computational modeling in geophysical fluid dynamics.
Professor Barreiro collaborates with both mathematicians and experimental neuroscientists at the
University of Washington and other institutions. Her work has appeared in journals such as
Physical Review E, Journal of Computational Neuroscience, and the
Proceedings of the Royal Society of London A.
Thomas W. Carr
Associate Professor (Ph.D. 1993, Northwestern)
Professor Carr's research focuses on the dynamics of physical systems modeled by nonlinear ordinary and partial differential euqations. He uses local and global bifurcation theory, asymptotic analysis, and numerical simulation and continuation to study the system's behavior and parameter sensitivities. Of particular interest is the synchronization characteristics of coupled oscillators. Areas of application include laser instabilities, coupled electronic circuits and mathematical biology.
He collaborates with scientists at the U.S. Naval Research Laboratory and the
Free University of Brussels, Belgium. His research has appeared in Physical
Review A, Physical Review E, Physical Letters A, Chaos, Physica D, and
SIAM Journal of Applied Mathematics.
Assistant Professor (Ph.D. 2002, University of California, Irvine)
Professor Chung's research activity has focused on nonlinear photonics, specifically, (1) analyzing both analytically and numerically the stochastic phenomena of electromagnetic field propagation in nonlinear random media, (2) numerical modeling of light propagation in photonic crystals, (3) optimizing and modeling of ultrashort high-power pulses in optical fibers. Currently, her research efforts involve a broader area, which includes (1) nonlinear dynamics of light in photonic crystals and nanophotonic devices; (2) error correction in noisy communication networks; and (3) developing efficient numerical method to model various aspects of highly distorted flow in fluid dynamics.
She has collaborated with the theoretical physics and experimental groups at Los Alamos National
Laboratory, and the Department of Mathematics at University of New Mexico, and the theoretical
physics group at Landau Institute in Russia. Her research has appeared in Journal of Nonlinear Science,
Optics Communications, Journal of Physics A: Mathematical and General, Physical Review E,
Optics Letters, Optics Express, and Nonlinearity.
Professor Emeritus (Ph.D. 1970, University of Manchester)
Professor Gladwell works in a variety of numerical analysis and scientific computation research areas - including ordinary differential equation initial and boundary value problems, mathematical software, and parallel computing - with an emphasis on developing tools to assist scientists and engineers with large-scale computing problems. His research has involved scientific computation for sintering and grooving of materials, diffusion-convection equations, waves generated by a semi-infinite plate, and chromosome synapsis in grasses.
Recently, he has worked on use of symbolic software for aiding the numerical solution of singularly perturbed boundary value ordinary differential equations, parallel codes for almost block diagonal systems, variable step Runge-Kutta-Nystrom algorithms for special second-order systems, integration of Hamiltonian systems, parallelization of numerical integration, and wavelet collocation for boundary value problems. He has been working with faculty members at the University of Manchester, Emory University and the Colorado School of Mines, with members of the Numerical Algorithms Group, and with former SMU graduate students at the Johns Hopkins University, Texas Instruments, Oakland University, the University of New Hampshire, the Polish Academy of Sciences, University of Texas at Dallas, Texas Women's University and Collin County Community College.
Professor Gladwell was a faculty member at the University of Manchester, UK, before
joining SMU in 1987. He has consulted with Texas Instruments and with NAG, is Editor-in-Chief
of the ACM Transactions on Mathematical Software, and an Associate Editor of the IMA
Journal on Numerical Analysis and of Scalable Computing: Practice and Experience, and
is the editor and author of seven research monographs and special journal issues.
His recent research has appeared in journals such as ACM Transactions on Mathematical
Software, Applied Numerical Methods, SIAM Review, Parallel Computing, Journal of
Computational and Applied Mathematics, Computational Materials Science, International
Journal of Numerical Methods in Engineering, Numerical Linear Algebra and its Applications,
Philosophical Magazine (Series A), and Computers and Mathematics with Applications,
Journal of Crystal Growth, and SIAM Review.
Professor (Ph.D. 1971, Massachusetts Institute of Technology)
Professor Haberman is the author of textbooks on ordinary and partial differential equations in science and engineering and on mathematical modeling in mechanical vibrations, population dynamics, and traffic flow. His research has involved various areas of physical applied mathematics: Hydrodynamic stability, solitons for nonlinear dispersive waves, slowly varying bifurcations in nonlinear ordinary differential equations, and transitions such as caustics and shocks in partial differential equations. He has investigated the change in action (adiabatic invariant) resulting from the slow passage of a separatrix and has extended this work to the slow passage of a separatrix for bifurcations of Hamiltonian systems. His work usually involves singular perturbation techniques: the methods of matched asymptotic expansions (boundary layers) and multiple scales (averaging).
Recently, he has used dynamical systems and singular perturbation methods to study chaotic collisions of nonlinear solitary wave in various different problems in fiber optics. He has analyzed the trapping of light due to interaction of a nonlinear pulse with an optical defect. He has introduced a universal separatrix map that explains the fractal structure in solitary wave interactions for dispersive nonlinear partial differential equations. He has been working with faculty members at the New Jersey Institute of Technology and the University of Vermont.
His research has appeared in the last few years in journals such as SIAM Journals of
Applied Mathematics and Applied Dynamical Systems, Physical Review Letters, Physica D,
Chaos, and the Journal of Nonlinear Science.
Professor (Ph.D. 1983, California Institute of Technology)
Prof. Hagstrom's research is focused on computational methods for simulating
time-domain wave propagation phenomena. Current projects include:
- The development and analysis of radiation boundary conditions and fast propagation algorithms for scattering problems utilizing novel plane wave representations of the wave field,
- Adaptive, high-order Hermite discretization methods on structured, composite or embedded grids,
- High-order/high-resolution discontinuous Galerkin discretizations on unstructured grids utilizing polynomial and nonpolynomial bases,
- Efficient time-stepping for equations with stiff components and on adapted grids.
Applications include electromagnetic and acoustic scattering, the generation of sound by unsteady and turbulent flows, gas-phase combustion, and the multiscale coupling of kinetic models, such as the Boltzmann equation, with continuum models such as the Navier-Stokes-Fourier system.
This research has been supported by the National Science Foundation, the Air
Force Office of Scientific Research, the Army Research Office, and NASA. Recent
publications have appeared in the SIAM Journals on Numerical Analysis,
Applied Mathematics, and Scientific Computing, the Journal of Computational
Physics, Communications in Applied Mathematics and Computational Science, the
Journal of Computational Mathematics, and Communications in Partial
Associate Professor, Emeritus (Ph.D. 1983, Technical Univeristy of Denmark)
Professor Melander's current research focuses on fundamental issues in vortex dynamics and statistical fluid mechanics. His topics include vortex/boundary interactions in 2-D, morphology of vortex interactions in 2- and 3-D, identification fo underlying mechanisms, topological description of 3-D viscous flows in terms of global bifurcation analysis of the vorticity field (i.e., vortex line history), construction and analysis of shell models of turbulence, statistical behavior of ensembles of shell model solutions, and the transition to turbulence in shell models.
Professor Melander's research is problem-driven and thus employs tools from
classical and applied mathematics, numerical analysis, and scientific
computation. Concepts from dynamical systems play a central role. His
publications have appeared in the Journal of Fluid Mechanics, Physics of
Fluids, Physical Review Letters, Fluid Dynamics Research, Physica D, and
Physical Review E.
Professor (Ph.D. 1988, Rensselaer Polytechnic Institute)
The primary focus of Professor Moore's research has been on developing adapative finite element methods for solving reaction-diffusion systems in one, two, and three space dimensions. Reaction-diffusion systems appear in a variety of applications. These include models in cardiac electrophysiology such as Fitzhugh-Nagumo, Ebihara-Johnston and Luo-Rudy I, combustion, catalytic surface reactions, and pattern formation. Adaptive methods have proved effective in solving such systems increasing the reliability, robustness and efficiency of standard methods.
Professor Moore was a faculty member of Tulane University for eleven years
before arriving at SMU. His published work has appeared in SIAM Journal on
Numerical Anaylsis, SIAM Journal of Scientific Computation, Journal of
Computational Physics, BIT, Physica D, Applied Numerical Mathematics,
Mathematical Biosciences and the Journal of Cardiovascual
Professor (Ph.D. 1983, California Institute of Technology)
Professor Reinelt's research involves mathematical modeling of applied physical problems and the development of solution techniques for these problems. Past problems he has examined include the motion of long bubbles in capillary tubes, fluid draining from a tube under the effect of gravity, and the shapes of bubbles and fingers as a fluid penetrates into a more viscous fluid.
His current research interests include foam flow, and the stability and variations of thin film thickness in coating flows. His foam research involves the development of theories that relate foam structure and the physical properties of the constituent phases and interfaces to macroscopic rheology. In the coating flow problem, he is investigating the instabilities that occur when a thin liquid film is deposited on a moving roller or cylinder. This problem is of particular interest to the printing and photographic industries, and other problems have applications in petroleum and geothermal energy production and manufacture of coatings and polymeric foams.
He recently has done research at the Laboratoire de Dynamique des Fluides Complexes at the Universite Louis Pasteur in Strasbourg, France, and has collaborated with colleagues in applied mathematics and fluid and thermal science at Sandia National Laboratories.
Some of his publications have appeared in the Journal of Fluid Mechanics,
Physics of Fluids, Journal of Colloid and Interface Science, and
International Journal of Multiphase Flow.
Assistant Professor (Ph.D. 2003, Rice University)
Professor Reynolds' research focuses on numerical methods of relevance to large scale scientific computing applications involving the nonlinear interaction of multiple physical processes, typically modeled using systems of partial differential equations (PDE). Such work aims to allow mathematical insight and innovation to impact the physical, biological and engineering sciences through the incorporation of increased realism into mathematical modeling systems, the development of increasingly robust and accurate numerical methods for solving these mathematical models, and the invention of computational algorithms to implement these numerical methods on increasingly large-scale computational hardware.
Specifically, Professor Reynolds investigates three fundamental applied mathematics issues: accurate modeling of physical systems involving disparate time and space scales, the development and use of highly-accurate and efficient time evolution algorithms for stiff multi-rate problems, and the investigation of discretization and solution methods that retain constraint-preserving properties of PDE models. To this end, Professor Reynolds relies on his expertise in large scale parallel computation as well as a broad range of numerical analysis techniques, including space-time discretization approaches for PDE systems and iterative solution approaches for nonlinear and linear systems of equations.
Professor Reynolds collaborates with scientists at the University of
California San Diego, Columbia University, SUNY Stony Brook, Lawrence
Livermore National Laboratory, Princeton Plasma Physics Laboratory, the
University of Neuchatel (Switzerland), and others. His published work
has appeared in SIAM Journal of Scientific Computing,
Journal of Computational Physics, Computational Methods
in Applied Mechanics and Engineering, Continuum Mechanics and
Thermodynamics, Systems and Control Letters, Future Generation Computer
Systems, Lecture Notes in Computer Science, ACM Transactions on
Mathematical Software, Journal of Physics: Conference Series,
Proceedings of SPIE, and Proceedings of ENUMATH.
Assistant Professor (Ph.D. 1998, TU Darmstadt, Germany)
Professor Rumpf's research covers topics in applied mathematics and in theoretical physics with an emphasis on statistical methods and on nonlinear dynamics. Focus of his scientific work has been the dynamics of spatially extended systems with applications in solid state physics, fluid mechanics, nonlinear optics, plasma physics and nanosystems. His special interest is the spontaneous formation of regular coherent structures from a turbulent or thermally disordered background.
Professor Rumpf collaborates with scientists from TU Chemnitz (Germany), Rensselaer Polytechnic Institute and the University of Arizona. His recent research has appeared in Physical Review Letters, Physical Review E, Europhysics Letters, Annual Reviews of Fluid Mechanics and Physica D, and he has edited a monograph on nonlinear dynamics of nanosystems.
Professor of Mathematics, Emeritus(Ph.D. 1964, California Institute of Technology)
Professor Shampine began working in numerical analysis just as it was emerging as a discipline and later helped start the subdiscipline of mathematical software. He joined SMU after a distinguished career at Sandia National Laboratories, where he was supervisor of the Numerical Mathematics Division. He is the author of several items of mathematical software used around the world and is a past president of the Association of Computer Machinery's (ACM's) Special Interest Group on Numerical Mathematics (SIGNUM). He has held numerous editorial positions and is an associate editor of the SIAM Journal on Numerical Analysis.
Much of his research has been directed toward more effective numerical solution of ordinary differential equations (ODEs). He is developing methods and software appropriate for graphical user interfaces. This work is being used for computer experiments in courses on ODEs and for scientific computation in the MATLAB environment.
He is the author of four books: Nonlinear Two-Point Boundary Value
Problems; Numerical Computing: An Introduction; Numerical Solution of Ordinary
Differential Equations: The initial Value Problem; and Numerical Solution
of Ordinary Differential Equations. His recent research has appeared in
journals such as Computers and Mathematics with Applications, SIAM Journal of
Numerical Analysis, SIAM Journal of Scientific and Statistical Computation,
Journal of Computational and Applied Mathematics, Applied Numerical Mathematics,
and Mathematics of Computation.
Assistant Professor (Ph.D. 2005, Virginia Tech University)
Professor Stigler's research focuses on the development of a mathematical framework for the reverse engineering of gene networks, using computational algebra as a primary source of tools. The main tool is that of Groebner bases of polynomial ideals. The models used in this work are time- and state-discrete finite dynamical systems, described by polynomial functions over a finite field. She and collaborators have developed an algorithm for reverse engineering gene networks from experimental time series data, including concentrations of mRNAs, proteins, and/or metabolite. She has applied this method to an oxidative stress response network in yeast and developmental networks in C. elegans and the fruit fly.
She has been actively involved with the DREAM (Dialogue for Reverse
Engineering Assessments and Methods) Initiative and SACNAS (Society for
Advancement of Chicanos and Native Americans in Science). One of her main
collaborations is with the Applied Discrete Mathematics Group at the
Virginia Bioinformatics Institute. Her work has been listed as one of the
top 25 Hottest Articles in Journal of Theoretical Biology.
Professor (Ph.D. 1995, Colorado State University)
Professor Tausch's research focuses on the numerical analysis of integral and partial differential equations. He has developed efficient numerical algorithms to solve problems that arise in Electromagnetics, Optics and Fluid Mechanics. He currently uses integral equation methods to solve high-dimensional parabolic equations that arise, for instance, in free surface or shape identification problems.
He has collaborated with engineers to simulate the behavior of integrated circuits, micro-mechanical devices and photonic waveguides. His software has been used in industry to help the design of semiconductor lasers with optical gratings.
His work has appeared in Mathematics as well as Engineering publications, such as, SIAM Journal of Scientific Computing, Mathematics of Computation, Computing, Journal of Numerical Mathematics, Journal of Computational Physics, Computational Mechanics, Journal of the Optical Society of America A, IEEE Transactions on Microwave Theory and Technology, IEEE Transactions on Computer-Aided Design and Inverse Problems.
Assistant Professor (Ph.D. 2002, Cornell University)
Professor Xu's research interests center on the development of computational techniques for problems in fluid mechanics and aerodynamics, including biological flows with tissues or membranes, supersonic and hypersonic turbulence with shockwaves, flow control by passive means, and fluid dynamics of nature's flyers and swimmers. The present development focuses on the immersed interface method, which models solids in a fluid with singular forces and solves the fluid flow subject to the singular forces by incorporating jump conditions into numerical schemes. The method is currently used to study the wing pitch reversal and fore-hind wing interaction in dragonfly flight.
Prior to joining the faculty at SMU, Professor Xu worked for GE Energy on steam turbine aerodynamics. He also worked at Cornell University and Princeton University as a post-doctoral research associate. His published work has appeared in Journal of Computational Physics, SIAM Journal on Scientific Computing, Physics of Fluids, and Journal of Fluid Mechanics.
Assistant Professor (Ph.D. 2002, Rice University)
Professor Zhou's research focuses on numerical linear algebra, scientific computing, and their broad range of applications; especially applications in material sciences and electrical engineering. He has developed algorithms that greatly improve the efficiency in solving large-scale eigenvalue problems arisen in density functional theory (DFT) calculations.
His current research includes extending the polynomial filtered subspace methods for generalized eigenvalue problems; improving/developing more efficient mixing-schemes for self-consistent field calculations; and extending the subspace techniques that have been successful for time-independent DFT to time-dependent DFT calculations.
His publications have appeared in Numerical Linear Algebra and Its Application, Physical Review Letters, Journal of Computational Physics, System and Control Letters, Journal of Applied Mathematics, and Computer Physics Communications.