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The bulk of the relentlessly-growing chaos literature consists of mathematical or experimental studies of particular systems or classes of systems. Professor Bradley is interested in moving beyond the analysis of chaos into the synthesis of chaos i.e. the active and intentional use of chaotic behavior to improve a system's design and performance.For example, sensitive dependence on initial conditions can be used to magnify the leverage of a small control action. The denseness with which state-space trajectories cover chaotic attractors can be exploited to widen the set of objectives that are reachable from a given starting point.
However, the same potentially-useful properties also give rise to a host of practical problems. Correspondences between models and systems--e.g., orbiting bodies, the double pendulum, nonlinear electronic oscillators, the Navier-Stokes equations--are inexact, and the differences become problematic in a highly-sensitive control scheme. Numerically-induced chaos, arising from the mechanical integrators that are used to simulate chaotic systems, is an equally difficult issue. Understanding of and solutions to both modeling and numerical problems are critical prerequisites to any scheme that is to exploit chaos in a practical way.
As our understanding of the natural world increases, the application of optimization techniques increases in two ways. Given a model of a physical or organization process, one can use optimization to select the parameters of the process to achieve the best outcome. In addition, many physical models may be understood by optimizing some key quantity such as potential energy, or least squares. Solving these optimization problems involves the design of sophisticated computer algorithms.In continuous optimization the problem is to minimize a real valued function of a set of variables possibly with some constraints involving equations or inequalities. Methods for doing this, such as quasi-Newton methods work fairly well for moderate size problems, but are not completely understood. As the number of variables increases, other methods must be used which take into account limits on the computer's ability to store data such as the limited memory methods Professor Byrd is working on. His work has application to solving potential energy minimization problems arising in molecular chemistry.
Current projects include: Nonlinear Domain Decomposition Methods, Numerical Studies of Bridge Performance, An Interactive Experimental/Numerical Simulation System with Applications in MEMS Design, Terascale Optimal PDE Simulations.
Professor Cary is interested in chaotic dynamics as a means for developing new dynamical approaches to understanding equilibrium, stability, and transport of plasma in three-dimensional configurations, plasma turbulence, and confinement of particles in accelerators.
Recently, his research has focused on calculating the velocity diffusion induced by turbulent electrostatic fields. For slowly growing plasma instabilities, standard textbook theories of "quasilinear diffusion" appear to be
Professor Cioffi-Revilla has authored numerous publications on the origins and evolution of political and international systems, including research on the causes of war and conditions of peace. His current research program, the Long-Range Analysis of War (LORANOW) Project, aims at developing a new theory with models and data sets on war and political processes in ancient international systems, using comparative analysis methods. In 1988, he received the Best Paper Award of the Symposium on Systems Engineering and Peace Research of the Austrian Society for Cybernetics, and the Citation for Outstanding Public Service on Behalf of the United Nations.
His latest book is Politics and Uncertainty: Theory, Models and Applications, Cambridge University Press (1998). His current book project is on "Origins of the International System: How Many Worlds Became One," proposing the new "pleogenic" theory of long-range political evolution based on computational modeling approaches.
Professor de Alwis' research is in theoretical particle physics. His main interests are supersymmetry, string theory and quantum gravity. In the last three years he has focused on quantum black holes (Hawking radiation etc.). Currently his is working on non-perturbative effects in supersymmetric field theories and in string theory. The mathematical techniques that are being used include group theory, differential geometry, algebraic geometry, and topology.
Professor DeGrand is a particle physicist interested mainly in the problem of quark confinement. There is a candidate theory for the strong interactions referred to as quantum chromodynamics (QCD); however, since quark confinement involves nonpertubative properties of QCD, it is very difficult to gain analytical understanding. His research methodology involves discretizing the equations of motion on a four dimensional space-time lattice and studying the theory via numerical simulation on large scale, supercomputers. Physics applications of Professor DeGrand's work include the spectroscopy of hadrons and the properties of their wave functions.
The mathematical tools this work requires, and on which research is conducted, include the development of algorithms for simulations of systems with many degrees of freedom and methods for the inversion of large, sparse, ill-conditioned matrices.
Mathematical Optimization is entering an era of stimulating cross fertilization with Artificial Intelligence. They are providing intriguing consequences for numerous disciplines ranging from business to biochemistry, and from psychology to environmental planning. New models and methods are changing long held conceptions about what is "solvable," and about what can properly be approached by the application of mathematical analysis. Advances in solution quality and in computational efficiency-which are fundamental to improving our ability to handle real world applications-are rapidly emerging in hard problem areas such as scheduling, optimal design, resource allocation, telecommunications, molecular structures, capital investment, pattern recognition and many others. Innovative methodologies based on metaphors of physical or mental processes (simulated annealing, genetic algorithms, neural networks, tabu search) are also changing our conception of the potential role of mathematical analysis in such areas, disclosing that intelligent design may transcend our current ability to establish theorem-proof relationships, and yet may provide a basis for formulating new relationships to be explored in the future.
Professor Goldman is a plasma theorist with keen interest in nonlinear waves and chaotic wave dynamics. He has studied nonlinear plasma wave- packet self-focusing and collapse in space physics applications. Such phenomena are excited by electron beams in the solar wind and planetary magnetospheres and by intense electromagnetic waves in Earth's ionosphere and in laser-irradiated pellets. Self-focusing and collapse are described by nonlinear Schroedinger equations and the so called "Zakharov equations." Closely related processes include parametric and modulational instabilities, wave phase conjugation, and other plasma wave interactions with counterparts familiar from nonlinear optics.
Many facets of contemporary research in hydrologic sciences involve topics from applied mathematics, including stochastic processes, which are also of contemporary mathematical interest and significance. For example, problems involving water and solute transport in porous media exhibiting multiple scales of spatial variability, and statistical-dynamic studies of paleoclimatic and paleohydrologic time series use the theory of Markov diffusions, stochastic differential equations and nonlinear dynamics. An important problem in this respect is to analyze the asymptotic behavior of Markov diffusions. This is equivalent to the study of asymptotics of the solutions of parabolic partial differential equations with space-dependent coefficients. Another area of Professor Gupta's research requires an understanding of the scale invariance/scale dependence in climate- vegetation-landscape-hydrologic interactions over a broad range of space-time scales. This area of research uses the newly developing mathematical theories of multifractal measures and asymptotics of spatial random networks.
Fluid mechanics is typically associated with the flow behavior of a conventional fluid, namely a gas or liquid. Nonetheless, the flow of solid particulates is wide-reaching, including applications in pharmaceuticals (mixing of pharmaceutical powders), chemical process industries (fluidized bed processing), environment (pollutant transport), astrophysics (planetary rings), and geophysics (landslides). Despite the prevalence of such flows, the mechanisms giving rise to puzzling behaviors, such as the de-mixing of unlike particles, are not well understood.To further this understanding, a variety of computational tools is used, ranging from ``molecular dynamics'' simulations of individual particles to continuum modeling of the entire solid phases. Although the computational requirements of the former are too high to be used for industrial-sized systems, it does provide an ``ideal experiment'' testbed for the latter, thereby giving way to descriptions which are both computationally feasible and fundamental in nature.
Professor Jessup's current research concerns algorithms and software for numerical solution of problems in linear algebra, specifically eigenvalue and singular value problems. Problems of this type arising in practical applications are often of very large order, and their large size makes finding accurate solutions both difficult and time consuming. Her work focuses on the design and implementation of algorithms for these problems that are fast on traditional uniprocessor computers or efficient on parallel computers while still ensuring accurate results.
Mathematical models for physical systems of inert or reactive fluids are formulated by using systematic methods to reduce general nondimensional differential equations to more elementary forms. The reduced equations are derived by considering special asymptotic limits based on the behavior of important nondimensional parameters. Perturbation methods are used to discover the significant time and length scales as well as to find the solution structure. Computational methods may be used to solve the reduced equations. This approach is being used to study the time history of combustion and explosion phenomena, including detonation wave initiation, the propagation of solidification fronts and coexisting rotational/acoustic processes driven by burning rate variations in models of solid rocket engines.
A large number of problems in practical settings are of a combinatorial nature. Some of these problems arise in the areas of planning and scheduling, telecommunications, and distribution systems. Modern heuristic techniques, based on artificial intelligence principles, offer the possibility of finding high quality solutions to these difficult problems. Professor Laguna's research explores the interface between operations research and artificial intelligence with the goal of designing improved solution procedures.
Some complex problems may often be described and formulated in terms of network structures. A related research interest consists of exploiting these structures through network flow programming techniques.
Numerical methods for electrical engineering design include continuous simulation of systems of nonlinear ODE's, discrete simulation of multivalued logic systems, multiple criterion optimization of signal processing systems and automatic synthesis of digital systems. These require techniques from multi-valued logic, temporal logic, process algebras, Petri nets, graph theory and combinatorial optimization.
The area of computer-aided design of integrated circuits has provided an extremely rich source of interesting mathematical and computational problems. Professor Lightner's research interests include both simulation and synthesis with the latter being the major focus. Automatic synthesis of digital systems has as its goal the production of a high quality (comparable to the best human designer) integrated circuit from an abstract, human readable description, in much less time than that required for humans.
Many of the major problems in synthesis are computationally intractable. Focus has been on a firm understanding of the theory, including the computational limitations, followed by the development of heuristic techniques for computing suboptimal solutions. The goal is to characterize the nature of the various approximate solutions, i.e. weak optimality conditions, and to develop systems to find these solutions.
Professor McBryan's interests include parallel computation, graphics and visualization, computational fluid dynamics, statistical mechanics and quantum field theory. A major focus of his current research is parallel supercomputers and their application to real-world applications such as aerodynamic design, oil reservoir simulation and weather modeling. Professor McBryan has used over a dozen parallel computer systems in his research and is currently developing heterogeneous systems in which disparate systems cooperate to complete massive computational tasks.
McBryan is a Director of CU's Center for Applied Parallel Processing (CAPP) which houses the Connection Machine CM-2, Kendall Square KSR-1 and Myrias SPS-2 parallel supercomputers.
Please see http://enstrophy.colorado.edu/~mohseni/ for more information.
- Nonparametric regression
- Thin-plate splines, neural networks, inference for function estimates, response surface methodology
- Time series
- Detection and properties of nonlinear systems, trend analysis
- Spatial statistics
- Spatial designs, Nonstationary processes
My research is in the area of kinetic theory and simulation of plasmas. Most of my current work is in the area of direct numerical simulation of tokamak plasma turbulence on large massively parallel computers. These simulations are done in a five dimensional phase space using newly developed particle-based methods. These calculations involve many millions of particles with self-consistently calculated electric fields. For the first time, these simulations are showing spectral features and transport levels at least qualitatively similar to turbulent transport observed in large present day experiments. Our goal is a fundamental understanding of plasma turbulence and transport in magnetized plasmas. Scientific visualization is utilized to analyze the features of the three dimensional turbulent fluctuations.
Fundamental research in magnetism has been in such diverse areas as domain wall dynamics in thin films, the effect of chemical short range and long range atomic order on the magnetic state of systems with competing ferromagnetic and antiferromagnetic order, spin canting in ferrites with nonmagnetic substitutions, microwave relaxation processes in ferromagnetic thin films, microwave loss mechanisms in ferrites, nonlinear dynamics in magnetic systems, magnetism in spin glasses, Brillouin light scattering on magnetic excitations, giant magnetoresistance in thin film sandwiches, and microwave magnetic envelope solitons in thin films.
Applied research has been concerned with studies of lunar soil magnetism, microstructure in ferrites, magnetic films for perpendicular recording and high density storage, metallic powders for absorber applications, new materials for millimeter wave applications, microwave soliton thin film devices, and surface damage effects in recording head materials.
Prof. Rutherford's research specialty involves the formulation and analysis of large-scale economic equilibrium models. He has made substantial and widely cited contributions to applied general-equilibrium analysis. His applied work ranges from an analysis of the effect of trade and economic growth to the economic effects of carbon emissions restrictions in response to global warming.
Professor Sani's research interests encompasses the analytical and numerical modeling of the fundamental aspects of engineering systems exhibiting coupled flow and transport. Current research focuses on the development of analytical and numerical methods appropriate for free and/or moving boundary problems. These occur, for example, in electrochemical plating or etching, coating flows and free surface flows often encountered in material processing. Specific questions on well-posedness in the continuum and the discrete Galerkin finite element method as well as the development of improved algorithms for the flow, transport and stability of such systems are being addressed.
Many problems in mathematical modeling and data analysis for scientific and engineering calculations lead to nonlinear optimization problems. Professor Schnabel's research is concerned with the solution of such problems, including unconstrained optimization, constrained optimization, systems of nonlinear equations, nonlinear least squares, and global optimization problems. This research includes the creation of new computational algorithms for these problems, the analysis of these algorithms, and the creation of mathematical software that implements these methods. Recently this research also is addressing applications of optimization to molecular chemistry.
A related research interest is parallel scientific computation. One aspect is the creation of parallel algorithms for solving optimization problems. A second aspect is creating effective programming languages and system support for parallel scientific computation on distributed memory multiprocessors.
Recent theoretical projects studied by Professor Shull include the physics of turbulent mixing layers in the interstellar medium, the thermal evolution of intergalactic plasma heated by ionizing radiation from quasi-stellar objects, the distribution and ionization structure of interstellar gas in the Milky Way disk and above the Galactic disk plane, the dynamical evolution of multiple supernova events produced by clusters of massive stars and the structure of radiative shock waves and supernova blast waves, including thermal conduction and dust-grain destruction.
In addition to theoretical programs, he is also involved with the analysis and modeling of data from NASA satellite telescopes, including the Hubble Space Telescope (HST), the International Ultraviolet Explorer (IUE), and the Infra-Red Astronomical Satellite (IRAS).
Professor Skodje's research centers on the dynamics of molecular processes, statistical dynamics of chemical reactions, intramolecular energy transfer, and collision phenomena. These issues are studied as problems in nonlinear dynamics. Area preserving maps along with a variety of techniques in dynamical systems theory are used to model physical problems. Topics in quantum chaos are of particular interest in modeling molecular phenomena where quantum mechanics may significantly alter mechanisms of energy flow and reactivity.
Professor Su's research interest is in the area of dynamical control and automation. He is currently conducting basic research for the development of technologies in three particular engineering applications. First, he is applying neural networks and fuzzy logic to modeling and control of computer wafer fabrication processes. Secondly, he is investigating new ways to advance the technology of high-authority autoflight systems. Thirdly, he is involved in research to advance the engineering capabilities in in-space construction of large space structures and spacecraft.
Professor Toomre's research centers on astrophysical fluid dynamics (AFD), with particular emphasis on nonlinear theories for compressible convection in stars. Numerical simulations on massively parallel computers serve as the major tools for studying such highly turbulent flows. Professor Toomre is active in helioseismology, using observations of the frequency- splitting of five minute oscillations of the sun to search for subphotospheric flows associated with giant cells and differential rotation in the convection zone of this star. Inverse theory has been developed to interpret the data. Such contact with observations provides challenges to three-dimensional numerical modelling of convection under strong rotational constraints.
Other theoretical work concerns thermohaline convection in the oceans producing stepped structures, nonlinear instabilities of shear flows, nonlinear dynamical systems arising from double-diffusive convection which exhibit multiple bifurcations and chaos, and internal gravity waves both in the solar atmosphere and in its deep core.
Many problems in engineering and science are very nonlinear in nature and have strong variabilities covering many intrinsic time and spatial scales. Numerical simulation of such problems requires either the use of highly adaptive numerical algorithms or the use of reduced models that capture ``important'' physics of the problem at a lower cost. Professor Vasilyev's work focuses on both creation of novel approaches for numerical simulation of complex multi-scale phenomena and development of low order ``physics-capturing'' models.
Professor Weidman's interests lie in the physics of fluid motion. Current areas of research include boundary layer flow, Stokes flow, water waves, vortex rings, capillary phenomena, and fluid instability.
Similarity reduction of partial differential equations play an important role in boundary layer theory. Flows over heated plates have different self- similar solutions near the leading edge and far downstream, and the conventional wisdom is that these flows must be matched to a non-similar flow between. Here we inquire whether distinct self-similar flows can be matched in a natural, perhaps globally self-similar, manner by invoking a more general similarity reduction ansatz.
Weakly and fully nonlinear solitary wave interactions are being studied in stationary and rotating systems. The concept of a soliton breaks down for waves of high amplitude and with it the idea that the waves incur a phase shift after interaction. The investigation here concerns the determination of a generalized time shift which describes interaction phase changes for nonlinear waves of both small and large amplitude.