Applied Mathematics - University at Buffalo

Applied Mathematics at UB

The University at Buffalo Mathematics Department has a tradition of leading research in Applied Mathematics. Our interests include

biological and stochastic modeling
bifurcation and stability theory
dynamical systems
material science and crystal growth
nonlinear waves and optics
numerical analysis
perturbation methods
population dynamics
scientific computation

Scientific computing plays an important role in many of these research projects. Several faculty have a network of SGI and Sun workstations with a 2-processor SGI Origin 200. The Math Department maintains a cluster of Sun workstations and PCs for general computing use and faculty have access to university-operated Sun and SGI timeshares. In addition, faculty have access to the Center for Computational Research (CCR) at UB which has cutting-edge supercomputers and visualization hardware.

The Applied Math group runs a weekly seminar series, with speakers from within the department and the unversity, as well as outside visitors. In some years, the seminars have been organized around a yearly topic. Example topics from past years include: Mathematical Finance, Thin Films, Phase Field Models, Stochastic Differential Equations, Crystallography.

In Spring 2006 the department hosted a workshop on Nonlinearity and Randomness in Complex Systems.

Faculty in Applied Mathematics

There are currently 11 faculty in the Applied Math group. Below are their areas of research, and several representative recent publications.

Gino Biondini - (Ph.D., University of Perugia, Italy) Nonlinear waves, integrable systems and their applications

The study of physical phenomena by means of mathematical models often leads to certain nonlinear partial differential equations which reveal a surprisingly rich mathematical structure. The study of these equations thus offers a unique combination of interesting mathematics and concrete physical/technological applications. Professor Biondini's research has two main goals: The first goal is to understand the properties of these equations and their solutions. This kind of research is usually called the study of "integrable systems", and requires a combination of techniques from different branches of mathematics. The second goal is to study the application of these nonlinear and/or stochastic systems to concrete physical situations, with the aim of obtaining results of practical usefulness. This often requires studying the combined effects of several kinds of perturbations which are often stochastic in nature, and can be done using exact methods, approximations (such as modeling, asymptotics and perturbative techniques), numerical methods (numerical modeling, Monte-Carlo simulations and variance reduction techniques) or combinations of all these approaches. Specific applications considered by Prof. Biondini are optical fiber communications, nonlinear optics and water waves.

Selected publications:

G Biondini, "Line soliton interactions of the Kadomtsev-Petviashvili equation", Phys. Rev. Lett. 99, 064103:1-4 (2007)

G Biondini, W L Kath and C R Menyuk, "Importance sampling for polarization-mode dispersion", J. Lightwave Technol. 14, 1201-1215 (2004). [Correction: J. Lightwave Technol. 24, 1065 (2006)]

J. Li, E. Spiller and G Biondini, "Noise-induced perturbations of dispersion-managed solitons", Phys. Rev. A 75, 053818:1-13 (2007)

G Biondini and Y. Kodama, "On the Whitham equations for the defocusing nonlinear Schroedinger equation with step initial data", J. Nonlin. Sci. 16, 435-481 (2006)

B. Prinari, M. J. Ablowitz and G. Biondini, "Inverse scattering transform for the vector nonlinear Schroedinger equation with nonvanishing boundary conditions", J. Math. Phys. 47, 063508:1-33 (2006)

Cliff Bloom - (Ph.D., NYU) Partial differential equations, scattering theory and energy methods.

Bloom CO, High frequency asymptotic solutions of the reduced wave equation on infinite regions with non-convex boundaries, Math Probl Eng 2, 333-365 (1996)
Bloom CO and Kazarinoff ND, Energy decay for hyperbolic systems of 2nd-order equations, J Math Anal Appl 132, 13-38 (1988)

Brian Hassard- (Ph.D., Cornell) Applied mathematics, bifurcation theory, precise numerical algorithms.

Du ZD, Hassard B, Precise computation of Hopf bifurcation and two applications, Dynamics of Continuous and Discrete Impulsive Systems A 8, 495-518 (2001).

Hassard B, An unusual Hopf bifurcation: The railway bogie, Int J Bifurcat Chaos 10 503-507 (2000)
Jeng JH, Hassard B, The critical wave number for the planar Benard problem is unique, Int J Nonlinear Mech 34 221-229 (1999)

Hassard BD, Counting roots of the characteristic equation for linear delay-differential systems, J Differ Equations 136 222-235 (1997)

Hassard B, Shiau LJ, A special point of Z(2)-codimension three Hopf bifurcation in the Hodgkin-Huxley model, Appl Math Lett 9 31-34 (1996)

Jae-Hun Jung- (Ph.D., Brown University) Numerical analysis and scientific computing, high order methods, spectral methods.

Jae-Hun Jung's main area of research is the numerical approximation of nonlinear hyperbolic conservation laws. Nonlinear hyperbolic conservation laws easily develop discontinuous solutions and the numerical approximation of such solutions is one of the most challenging problems in the area of numerical analysis. This research aims to develop stable and efficient numerical methods and apply them to various physical and engineering problems. His current research topics include the following: Multi-domain spectral penalty methods for multiscale problems, High order hybrid methods, Radial basis function methods, Resolution of the Gibbs phenomenon, Spectral approximations of numerical relativity, Computational fluid dynamics, Kinetic theory, Hybrid methods for magnetized PEM fuel cells.

Recent publications:

S. Gottlieb and J.-H. Jung, Numerical issues in the implementation of high order polynomial multi-domain penalty spectral Galerkin methods for hyperbolic conservation laws, Communications in Computational Physics, 5 (2-4), 600-619, 2009.

W. S. Don, D. Gottlieb and J.-H. Jung, A weighted multi-domain spectral penalty method with inhomogeneous grid for supersonic injective cavity flows, Communications in Computational Physics, in press, 2008.

J.-H. Jung and B. D. Shizgal, On the numerical convergence with the inverse polynomial reconstruction method for the resolution of the Gibbs phenomenon, Journal of Computational Physics, Vol. 224, pp. 477-488, 2007.

J.-H. Jung, A note on the Gibbs phenomenon with multiquadric radial basis functions, Applied Numerical Mathematics, Vol. 57, pp.213-239, 2007.

J. Rosen, J.-H. Jung and G. Khanna, Instabilities in numerical loop quantum cosmology, Classical and Quantum Gravity, Vol. 23, pp. 7075-7084, 2006.

Avner Peleg - (Ph.D., The Hebrew University of Jerusalem, Israel) Applied mathematics, nonlinear waves, pattern formation, stochastic processes.

Dr. Peleg's research interests are in applied mathematics of optical communications (in optical fibers and in the atmosphere), and in mathematical modeling of dynamics of phase transitions (with applications in materials science). The primary research effort concerns propagation of pulses of light in multichannel optical fiber communication systems. In multichannel transmission many pulse sequences propagate through the same optical fiber and as a result, collisions between pulses from different sequences (corresponding to different channels) are very frequent and can lead to severe limitation on system performance. Since state-of-the-art multichannel systems use more than 100 channels and since the pulse sequences are random, the task of obtaining an accurate description of the dynamics is a very challenging one. One nonlinear effect that is particularly important in these systems is called delayed Raman response. One of the goals of this research is to understand the ways in which the interplay between delayed Raman response and other random and nonlinear processes affects pulse propagation and transmission quality. Another goal is to find relations between the dynamical behavior of the optical pulses and the dynamics of coherent patterns in other fields, such as pattern formation and turbulence.

Other research topics include:
(1) Propagation of multiple laser beams in atmospheric turbulence;
(2) Modeling of interface-controlled relaxation dynamics of two-phase systems.

Recent publications:

A. Peleg, Y. Chung, T. Dohnal, and Q.M. Nguyen, Diverging probability density functions for flat-top solitary waves, Phys. Rev. E, Vol. 80, 026602 (2009).

Y. Chung and A. Peleg, Monte Carlo simulations of pulse propagation in massive multichannel optical fiber communication systems, Phys. Rev. A, Vol. 77, 063835 (2008).

A. Peleg, Intermittent dynamics, strong correlations, and bit-error-rate in multichannel optical fiber communication systems, Phys. Lett. A, Vol. 360, 533-538 (2007).

P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J.V. Moloney, Optimized multi-emitter beams for free-space optical communications through turbulent atmosphere, Opt. Lett., Vol. 32, 885-887 (2007).

M. Conti, B. Meerson, A. Peleg, and P.V. Sasorov, Phase ordering with a global conservation law: Ostwald ripening and coalescence, Phys. Rev. E, Vol. 65, 046117 (2002).

Bruce Pitman - (Ph.D., Duke) Applied mathematics, scientific computing.

Professor Pitman works in two distinct areas. One is granularmaterials. Much of this work involves large-scale geophysical mass flows -- for example debris flows, pyroclastic flows, and mudslides. Together with colleagues in Geology, Geography and Engineering, Pitman has begun a program of research involving mathematical modeling, high performance computing, and statistical and stochastic analysis of uncertainty, with the goal of better understanding the hazard risk posed by these flows. A new project with colleagues at Duke, Marquette, Valencia and SAMSI is directed at quantifying the risk and uncertainty of uncertain events such as volcanic mass flows. These models and simulation capabilities, together with visualization and communication tools, will be used to assist scientists and public safety officials in hazard risk assessment.

The second of Pitman's research interests is renal hemodynamics. Together with colleagues in math and physiology, Pitman has been studying the dynamics of the Tubuloglomerular feedback system (TGF). TGF acts to regulate blood flow into the nephrons, the primary functional unit in the kidney. It is in the nephrons that blood is filtered and ultimately concentrated into urine. Bringing together techniques of applied mathematics and advanced scientific computing, the models developed by this group offer insight into the bifurcation phenomenon and oscillatory flow measurements seen in experimental work.

Recent papers:

"Parallel adaptive numerical simulation of dry avalanches over natural terrain" Journal of Volcanology and Geothermal Research 139 (2005) 1- 21. This paper describes the TITAN2D simulation environment for mass flows, and provides comparisons of simulation results and experimental data. (pdf)

"A two-fluid model for avalanche and debris flows" Phil. Trans. R. Soc. A (2005) 363, 1573-1601. This paper presents a depth-averaged model of 2 phase flows, appropriate for liquid-rich debris flows and mud flows.

"A Reduced Model for Nephron Flow Dynamics Mediated bt Tubuloglomerular Feedback" in Membrane Transport and Renal Physiology H.E. Layton and A.M. Weinstein (eds.) Springer Verlag (2002) p. 345-364. This paper presents a delay-integral model of the TGF system and compares Hopf bifurcation results with our well-established PDE model.

Jim Reineck - (Ph.D., Wisconsin-Madison) Dynamical systems, Conley Index.

Mrozek M, Reineck JF, Srzednicki R, The Conley index over a base, T Am Math Soc 352 4171-4194 (2000)
Mrozek M, Reineck JF, Srzednicki R, The Conley index over the circle, J. Dynam. Differential Equations 12 385-409 (2000)

Mischaikow K, Mrozek M, Reineck JF, Singular index pairs, J. Dynam. Differential Equations 11 399-425 (1999)

Gedeon T, Kokubu H, Mischaikow K, Oka H, Reineck JF, The Conley index for fast-slow systems. I, One-dimensional slow variable, J. Dynam. Differential Equations 11 427-470 (1999)

Reineck JF, The connection matrix in Morse-Smale Flows. 2., T Am Math Soc 347 2097-2110 (1995)

John Ringland - (Ph.D., Texas-Austin) Applied mathematics, bifurcation theory, mathematical modeling, computational mathematics.

Professor Ringland works in the qualitative and quantitative analysis of deterministic and stochastic dynamical systems:

Mohammed-Awel J, Kopecky K, Ringland J, A situation in which a local nontoxic refuge promotes pest resistance to toxic crops, Theor. Pop. Biol. 71, 131-146 (2007)

Brucks K, Ringland J, Tresser C, An embedding of the Farey web in the parameter space of simple families of circle maps, Physica D 16, 142-162 (2002)

Brian Spencer - (Ph.D., Northwestern) Materials modelling, free boundary problems, instabilities and pattern formation.

Professor Spencer's research interests are in the applied mathematics of materials. The research combines physics-based mathematical models with asymptotic, analytic and numerical methods to describe growth processes, instabilities and microstructure formation in materials. Specific research programs include:

instabilities and pattern formation in strained alloy film deposition
formation of quantum dots in strained solid films
corner regularizations in crystal growth models