Faculty Research Descriptions
Alejandro Aceves
Professor
Ph.D., University of Arizona 1988
Professor Aceves' research focuses on the mathematical modeling of phenomena in nonlinear optics. By using techniques from perturbation theory and asymptotic's 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 Ph.D. 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.
Vladimir Ajaev
ajaev@smu.edu | Website |
Professor
Ph.D., Northwestern, 1999
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 Aix-Marseille, France, the Technical University of Darmstadt, Germany, and the Institute of Thermophysics in Novosibirsk, Russia. 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 Interface Science.
Andrea Barreiro
Associate Professor
Ph.D., New York University, 2006
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.
Difeng Cai
Assistant Professor
Ph.D., Purdue University, 2019
Professor Cai’s research interest focuses on efficient algorithms for numerical linear algebra, partial differential equations and integral equations. Application areas include physics, statistics, machine learning and deep learning. Professor Cai's research aims to resolve grand computational challenges in scientific computing and data science by studying the mathematical properties and designing efficient algorithms. Examples include modern matrix techniques for massive data analysis, adaptive solution for partial differential equations, etc. His recent interests involve studying mathematical structures in deep learning models for scalable computation and using deep learning approaches such as generative models for solving challenging scientific computing problems.
Professor Cai's work has appeared in Numerical Linear Algebra and Its Application, Numerische Mathematik, Journal of Computational Physics, SIAM Journal on Scientific Computing, SIAM Journal on Matrix Analysis and Applications, IEEE International Parallel and Distributed Processing Symposium, Conference on Uncertainty in Artificial Intelligence, and elsewhere.
Wei Cai
Clements Chair Professor
Ph.D., Brown University, 1989
Professor Cai’s research interest focuses on developing advanced fast machine learning, stochastic, and deterministic numerical methods to understand complex physical phenomena with a special focus on quantum phenomena, electromagnetic processes, and wave scattering in random media. He has published over 130 refereed articles in top international journals such as SIAM J. on Numerical Analysis, SIAM J. on Scientific computing, Computer Physics Communications, Journal of Computational Physics, IEEE transactions on MTT, among others, and is the author of the book "Computational Methods for Electromagnetic Phenomena: electrostatics in solvation, scattering, and electron transport" published by Cambridge University Press, 2013. He serves on the editorial boards for Communications in Computational Physics, Journal of Computational Mathematics, and Communications in Mathematics & Statistics. His research has been funded by the National Science foundation, Department of Energy, US Army Research Office, the National Institute of Health, and US Air Force Office of Scientific Research.
Thomas Carr
tcarr@smu.edu | Website |
Associate Professor
Ph.D., Northwestern, 1993
Professor Carr's research focuses on the dynamics of physical systems modeled by nonlinear ordinary and partial differential equations. 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.
Mohamad Kazem Shirani Faradonbeh
mohamadksf@smu.edu | Faculty Site
Assistant Professor
Ph.D., University of Michigan, 2017
Dr. Faradonbeh's research focuses on different aspects of algorithms that can bridge from limited data to reliable outcomes, yet enjoying computational friendliness. That includes three major categories focusing on design and analysis of algorithms, with applications in precision medicine, healthcare management, online advertisement, intelligent tutoring, precision agriculture, and system engineering.
The first is the study of methods that can accurately learn many parameters from structured data, as well as experimentation for collecting useful data. Commonly, different technical tools from high-dimensional statistics and Bayesian machine learning are utilized to address issues such as spatiotemporal dependence and compound uncertainties.
The second category is that of optimal decision-making in settings such as network systems and patient scheduling. The important issues include NP-hardness and guaranteed reliability, and approaches from combinatorial optimization and stochastic control are used.
Projects in the third category consist of reinforcement learning algorithms for real-time data-driven prescriptions, e.g., for tutor recommendation or dynamic treatments. Along these axes, fundamentally applicable settings are adopted together with a variety of frameworks from online learning and stochastic analysis, among others.
Weihua Geng
Associate Professor
Ph.D., Michigan State University, 2008
Professor Geng's research focuses on modeling problems in structural and systems biology using differential equations and integral equations, and solving these problems numerically with fast algorithms and high performance computing. Particularly, Professor Geng is interested in numerically solving interface problems described by PDEs and integral equations. These problems appear when adjacent layers with different physical, chemical, and biological properties are present. As far as the numerical algorithms are concerned, Professor Geng is interested in 1) matched interface and boundary (MIB) methods (high order, finite difference mesh-based algorithms used in solving elliptic PDEs with discontinuities and singularities); and 2) boundary integral methods combined with treecode (a fast algorithm for evaluating interactions of N-body problems) to achieve high accuracy and efficiency. Recently Professor Geng has been working on the extension of these numerical algorithms to electrodynamics and fluid mechanics for the structural biology study at the molecular level.
Professor Geng collaborates with mathematicians, computer scientists, and biologists at the Michigan State University, University of Michigan, University of Alabama, Pacific Northwest National Laboratory, and other institutions. His work has appeared in journals such as Journal of Computational Physics, Journal of Computational Chemistry, Computer Physics Communications, and Journal of Chemical Physics.
Thomas Hagstrom
Professor
Ph.D., California Institute of Technology, 1983
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 discretization on
unstructured grids utilizing polynomial and non-polynomial 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 multi scale 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 Differential Equations.
Barry Lee
Associate Professor
Ph.D., University of Colorado at Boulder, 1996
Professor Lee’s research focuses on the mathematical modeling, numerical algorithmic development, and scientific computing of large-scale industrial and laboratory applications. His current research interests include efficient methods for the Boltzmann transport equation (neutron/photon transport), Maxwell equations (fusion), equations of elasticity (structural designs), general coupled systems of elliptic partial differential equations (uncertainty quantification), and large systems of algebraic-differential equations (electric power grid networks). Central to his research is the development of schemes that give optimal computational efficiency on serial and large-scale parallel computer platforms. Thus, an essential component of his research is computational linear algebra, particularly scalable multigrid and multilevel methods.
His current collaborations include mathematicians and physicists at the Lawrence Livermore, Pacific Northwest, and Argonne National Laboratories. His research has appeared in SIAM J. Scientific Computing, SIAM J. Numerical Analysis, Numerical Linear Algebra with Applications, Mathematics of Computation, and Electronic Transactions of Numerical Analysis. For the past 15 years, he has been affiliated with several Department of Energy national laboratories.
Peter Moore
Professor
Ph.D., Rensselaer Polytechnic University, 1988
The primary focus of Professor Moore's research has been on developing adaptive 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 51°µÍø. 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 Cardiovascular Electrophysiology.
Scott Norris, Ph.D.
Associate Professor
Northwestern, 2006
Professor Norris's research focuses on multi-scale continuum descriptions of problems at small scales in materials science, with an emphasis on thin films dominated by interfacial phenomena. This involves a variety of mathematical techniques, including continuum modeling, asymptotic and perturbation methods, linear and nonlinear stability analysis, various kinds of numerical simulation, and the extraction of meaningful statistics from large data sets. Recent specific topics of interest include the coarsening of faceted crystal films, spontaneous nanoscale pattern formation on ion-irradiated semiconductors, and the growth of nanostructured solids by means of phase separation during deposition.
Professor Norris collaborates with scientists at Harvard University, the University of Helsinki (Finland), the University of Glasgow (Scotland), the Helmholtz Center of Dresden-Rossendorf (Germany), the Ecole Polytechnique (France), and the University of Kentucky. His publications have appeared in Nature Communications, Physical Review Letters, Physical Review B and E, Journal of Applied Physics, Journal of Computational Physics, Acta Materialia, Journal of Crystal Growth, and Nuclear Instruments and Methods in Physics Research B.
Daniel Reynolds
Professor
Ph.D., Rice University, 2003
Applied Mathematics pertaining to Scientific Computation (large scale parallel, multi-physics, nonlinear, PDE systems) and Numerical Analysis (multirate time integration, iterative linear and nonlinear solvers). Application areas include climate, plasma physics, cosmology, and materials science.
In recent decades computation has rapidly assumed its role alongside theory and experiment as the third pillar of the scientific method. During this time, the complexity of scientific simulations has evolved from relatively simplistic calculations involving only a handful of basic equations, to massive models that combine vast arrays of physical processes. While early simulation approaches could be well-understood using standard mathematical techniques, applied mathematics has unfortunately not kept up with the fast pace of scientific simulation development. The result of these differing paths is that many numerical analysts spend their effort constructing elegant solvers for simple models of little practical use, while computational scientists study highly-realistic systems but "solve"' them using ad hoc methods with questionable accuracy and stability.
Professor Reynolds' research aims to bridge this gap between mathematical theory and scientific computing practice, through the creation, application, and dissemination of flexible algorithms that may be tuned for modern, multiphysics problems, while still providing mathematical assurances such as accuracy, stability and parallel scalability.
Professor Reynolds collaborates with a wide range of scientists on applications that include fusion energy, climate modeling, numerical weather prediction, cosmology, and additive manufacturing, with research funding from the U.S. Department of Energy, the U.S. National Science Foundation, and the U.S. Department of Defense. His published work has appeared in SIAM Journal of Scientific Computing, Journal of Computational Physics, ACM Transactions on Mathematical Software, Computer Physics Communications, Journal of Advances in Modeling Earth Systems, Geoscientific Model Development, Modelling and Simulation in Materials Science and Engineering, The Astrophysical Journal, International Journal of High Performance Computing, and elsewhere.
Benno Rumpf
Associate Professor
Ph.D., TU Darmstadt, Germany, 1998
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.
Brandilyn Stigler
Associate Professor
Ph.D., Virginia Tech University, 2005
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.
Johannes Tausch
Professor
Ph.D., Colorado State University, 1995
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.
Sheng Xu
sxu@smu.edu | Website |
Associate Professor
Ph.D., Cornell University, 2002
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 shock waves, 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 51°µÍø, 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.
Mikhail Zaslavsky
mzaslavskiy@smu.edu | Faculty Site
Assistant Professor
Ph.D., Lomonosov Moscow State University, 2004
Professor Zaslavsky's scientific interests mostly lie in the fields of model reduction, linear algebra, approximation theory, multi-scale methods, homogenization with application to solution of forward and inverse PDE problems. While most of the numerical approaches he's developed were primarily targeted to geophysical exploration, he is constantly looking for other areas of applications. In particular, dueto much in common in the nature of measurements, he has started exploring applications in radar imaging, medical imaging as well propellant tank imaging for NASA applications. Also, he is currently investigating the natural extension of his developments to the area of data science and unsupervised machine learning. A wide range of applications here includes bioinformatics, geophysics, social network studies, financial mathematics, etc.
Prof. Zaslavsky strongly believes that a good sign of novel theoretical development is that it was originally driven by application in mind. And, vice versa, it is unlikely to get a breakthrough in applications without new solid theory. Hence, he constantly tries to keep tight connections with his current collaborators from both academia and industry as well as to explore new areas and ideas for research.
Yunkai Zhou
yzhou@smu.edu | Website |
Associate Professor
Ph.D., Rice University, 2002
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.