Research
Applied and Computational Mathematics
The diverse research interests among the members of this group have a unified theme-namely, the development of mathematical modeling and computational techniques for the study of nonlinear phenomena.
Raymond Chin
Prof. Chin's research is motivated by the need to develop accurate and efficient algorithms to compute solutions to multiple scale problems in science and engineering.These methods are designed to take advantage of the inherent mathematical and physical properties of the underlying problem. Hence, a combination of asymptotical and numerical techniques is employed in their development. Domain decomposition methods reminiscent of matched asymptotic methods are natural vehicles to convey these hybrid techniques. He has developed Gaussian quadrature methods for an exponential weight, and this includes the numerically difficult problem of generating the recurrence coefficients of the associated orthogonal polynomial given a weight function. The intended application is toward modifying methods for integrating stiff ordinary differential equations inherent in problems of chemically reacting type modeling enzyme-substrate interaction, pharmacokinetic reactions, electrical networks, etc. In turn, the modified ODE solvers are used in system identification or parameter estimation problems associated with phenomenological modeling of complex bio-physio-chemical systems of medical interest.
Alexey Kuznetsov
Prof. Kuznetsov's central professional goal is to establish interdisciplinary research that combines experimental and theoretical biology with mathematical and statistical modeling. He is an applied mathematician by training. His dissertation addressed phenomena of synchronization and pattern formation in active, oscillatory networks using qualitative theory of differential equations and computational modeling. In addition to mathematics, He learned biophysics and molecular biology during my postdoctoral training. This cross-disciplinary background has proven to be a significant advantage in seeking explanations to biological problems via mathematical and computational modeling. In the future, Alexey wants to connect different levels of biological research, from molecular and cellular to system and behavioral, focusing on medical and pharmaceutical problems and further developing his experimental and medical collaborations.
Bart Ng
Prof. Ng has recently returned to the study, using asymptotic tech niques, of the spectra of the classical Orr-Sommerfeld problem which governs the linear stability of parallel shear flows. In contrast to his earlier work in this area which was concerned primarily with the derivation of uniform asymptotic approximations, he re-examine the problem of obtaining "first-order" approximations to the eigenvalue relations. By relaxing the uniformity requirement, he shows that it is relatively straight forward to derive simple outer approximations that are complete in the sense of Olver and remarkably, these approximations involve only elementary transcen dental functions. Despite their striking simplicity, however, these new results provide for the first time a global analytical description of the spectra of a large class of benchmark hydrodynamics stability problems. For example, these approximations have made it possible to compute the eigenvalues of semi-bounded flows that lie arbitrarily close to certain continuous spectra where all previous treatments (analytically as well as numerical) have failed. It appears that the resulting techniques can easily be adapted to deal with a broad class of higher-order turning-point problems of hydrodynamic type. Presently, they are also exploring the applicability of these new techniques to the study of the spectra of the non self-adjoint Zakharov-Shabat eigenvalue problem that has played a role in the derivation of a number of canonical equations of mathematical physics with "soliton" properties.
Leonid Rubchinsky
Prof. Rubchinsky's research lies in the area of mathematical and computational neuroscience; in particular, in applications of dynamical systems to problems of neurobiology and medicine. He employs both mathematical and computational methods to study the dynamics of the nervous system to gain insights into its function. Mathematical models, in particular, help to bridge the gap between different levels of biological knowledge, provide insights into the underlying principle governing the function of neuronal system and, in turn, are an effective tool for developing practical applications (such as treatment strategies). Rubchinsky's current research centers on the dynamics of basal ganglia-brain nuclei, which, among other things, control motor programs and impact on Parkinson's disease. He developed conductance-based models to study dynamics of motor control in basal ganglia networks. These models allow for study of how changes in cellular properties affect the behavior of healthy and Parkinsonian brains. From clinical recordings obtained from brains of Parkinsonian patients and mathematical methods developed for the quantitative characterization of the short-lived synchronization, Rubchinsky and coworkers offer hypotheses on the structure and dynamical functioning of basal ganglia motor control networks that are responsible for tremor generation. It appears that basal ganglia networks can be described by the networks of coupled oscillators near the threshold of instability of synchronization manifold. Rubchinsky collaborates with Karen Signart (University of California, Davis) and Nancy Kopell (Boston University) on the development of dynamical models of basal ganglia function. He initiated a collaboration with colleagues from IU Medical School and from Krasnow Institute at George Mason University to study the oscillatory dynamics of therapeutic anti-parkinsonian deep brain stimulation.
Dan Rusu
Prof. Rusu's research is in the interdisciplinary area of pattern formation in physical systems, which combines theoretical, applied and computational aspects. Pattern formation is observed in a wide variety of contexts and there is a growing realization that certain aspects are independent of physical details of the particular model and largely determined by purely mathematical properties such as geometry and symmetry. The general context for a symmetry based analysis of pattern formation is equivariant bifurcation theory. In particular, the role of circle symmetry as a theoretical framework for studying dissimilar two-dimensional annular convections such as those from geophysics and electrohydrodynamics is investigated. The two convection problems occur on vastly different spatial scales and differ physically in the source of the instability, while sharing common mathematical features. Both are especially interesting because they provide physically relevant examples of two-dimensional Navier-Stokes flows. Currently, the analytical and computational tools that Prof. Rusu developed for studying spatio-temporal pattern formation in thermoconvection are extended to other convective phenomena. The inclusion of various types of convections in a single theoretical framework supports the model independency approach idea in physics. The practical significance of this research lies in the fact that it captures the essential phenomenology and provides a systematic description with less computational effort than a direct numerical simulation of the partial differential equations models.Asok Sen
Professor Sen's research is in the area of physical applied mathematics. He uses advanced signal processing techniques to analyze various types of oscillatory activity in biomedical, engineering and environmental systems. In particular, his research involves application of wavelet analysis to delineate the dominant oscillatory modes and describe how these modes vary with time. An advantage of the wavelet-based approach is that it can be applied to nonstationary processes. A specific aim of his research is to characterize intermittent phenomena that occur frequently in many of these systems. The intermittent dynamics is also analyzed by means of statistical methods and multifractal approaches. These techniques are being applied to the analysis of (a) local field potentials recorded from the subthalamic nucleus of patients with Parkinson's disease, (b) EEG records from epileptic patients and kindled epileptic rats, (c) functional MRI data of hand movement, (d) riverflow and lake-level time series, (e) paleoclimatic records from the ocean, (f) stalagmite records from caves, (g) pressure and heat release fluctuations in internal combustion engines, (h) dynamics of cracked rotors, and (i) solar activity.
Luoding Zhu
Prof. Zhu research program is in the area of applied computational mathematics with biological applications. He is interested in numerical methods for and computer simulations of fundamental mechanical and/or biological processes which involve incompressible viscous fluids and elastic deformable boundaries. There are two major components of his research program: development of numerical methods for fluid-flexible-structure-interaction problems including extension/improvement of the immersed boundary (IB) method, and applications of these methods to problems in life science/ biomedical engineering.
Currently Zhu works on 1) developing a 3D implicit IB method using the lattice Boltzmann approach with applications in problems of biomedical interest -- blood flows with transport and reacting constituents interacting with a compliant vessel wall covered by an endothelial surface layer underlie the initiation and development of atherosclerosis; and the interaction of air flow through the pharyngeal airway with the genioglossal muscle, a process involved in sleep apnea, i.e. a disorder characterized by repetitive collapse of the pharyngeal air way during sleep. 2) developing novel numerical methods for modeling and simulations of red blood cells interacting with flowing blood using the thin-shell theory and the Navier-Stokes equations.