Department of Mathematics

Applied and Interdisciplinary Mathematics Seminar

Winter 2000

The seminar meets on Friday, 3pm , in 4096 East Hall unless otherwise noted.



Questions? Comments? Please email karni@math.lsa.umich.edu .



Friday, Jan 7
Mark Alber, University of Notre Dame
"Piecewise Wave Solutions of the Nonlinear Evolution Equations"

Piecewise modeling is widely used in the theory of nonlinear dynamical systems with applications in engineering and biology. Recently peakon type (piecewise wave) solutions of the evolution equations became the subject of an active research effort.

In this talk we demonstrate some special features of the integrable evolution equations admitting piecewise solutions that differentiate these equations from those whose solutions are smooth, such as the KdV equation.

Then we describe a method of constructing weak piecewise wave solutions of such equations consisting of different pieces of profiles glued together at the peak points and describe propagation of the peaks in the context of the level set theory.

  • Friday, Jan 14
    Asen Dontchev, Mathematical Reviews, AMS
    "Newton Method for Convex Best Approximation"

    We consider the problem of finding a convex function which interpolates given points and has a minimal $L^2$ norm of the second derivative. This problem reduces to a system of equations involving semismooth functions. We study a Newton-type method and prove that its local convergence is superlinear, thus settling a question raised by Micchelli et al. (1985).

  • Friday, Jan 21
    Amnon Meir, Auburn University
    "Finite Element Simulation of MHD Flows, with an Application to Liquid Metal Processing"

    Magnetohydrodynamics (MHD) is the theory of the macroscopic interaction of electrically conducting fluids with magnetic fields. It is of importance in geophysics and in connection with many engineering problems such as plasma confinement, electromagnetic casting, and liquid-metal cooling of nuclear reactors.

    In the viscous incompressible case, MHD flow is governed by the Navier-Stokes equations and Maxwell's equations of the magnetic field. In this talk I will describe a novel approach (I developed with P. G. Schmidt) to the mathematical analysis and finite-element simulation of such flows, discuss some results regarding the existence and uniqueness of solutions and the convergence of approximation schemes, and show applications to several MHD flow problems arising in the electromagnetic processing of liquid metals.

  • Monday, Jan 24
    Oscar Gonzalez, Swiss Federal Institute of Technology
    "Energy and momentum conserving algorithms in continuum mechanics"

    Deformable continuum bodies exhibiting elastic response have applications in many diverse areas of physics, chemistry, biology and engineering. For example, in the study of biomolecules and polymers, elastic-body models offer a simplified means of calculating structural or dynamical properties of what would otherwise be very large-dimensional atomic systems. In this talk I address the issue of designing stable, robust time-stepping schemes for fully nonlinear systems in classical and continuum mechanics that are faithful to the energetic and geometric properties of the system. Standard schemes like the implicit mid-point rule generally fail in this respect due to a numerically-induced coupling between rotations and internal deformations. I show that this coupling leads to a provable instability in a simple finite-dimensional system, and that this instability also arises in complex systems such as nonlinear elastodynamics. By modifying the mid-point rule in a way that respects rotational symmetry, one can regain stability, and arrive at schemes that conserve energy and momentum related integrals. I show these conserving schemes can be understood in terms of the concept of a discrete gradient. In particular, discrete gradients lead to schemes that faithfully capture the energetic and geometric properties of a system, even in the presence of physical damping.

  • Tuesday, Jan 25
    Oscar Gonzalez, Swiss Federal Institute of Technology
    "Global Curvature, Ideal Knots and Models of DNA Self-Contact"

    Experiments on knotted DNA molecules suggest that certain physical properties of DNA knots can be predicted from a corresponding ideal shape. Intuitively, when a given knot in a piece of string is pulled tight, it always achieves roughly the same geometrical configuration, with a minimum length of string within the knot. Such a configuration is called an ideal shape for the knot, and approximations of ideal shapes in this sense have been found via a series of computer experiments. These shapes have intriguing physical features and have been shown to capture average properties of knotted polymers. But when does a shape satisfy the intuitive geometrical definition for ideality? In this talk I show that ideal shapes can be understood using only elementary (but new!) mathematics. In particular, I show that global curvature, a very natural and simple generalization of the classic concept of local curvature, leads to a simple characterization of an ideal shape and to a necessary condition for ideality. Another application of global curvature can be found in characterizing the equilibria of knotted curves or rods, which may exhibit self-contact after sufficient twisting. Here global curvature provides a simple way to formulate the constraint that prevents a rod from passing through itself.
  • Thursday, Jan 27
    Tobin Driscoll, University of Colorado
    "High-order time stepping for linearly stiff nonlinear propagation equations"

    Many nonlinear wave equations of theoretical and practical interest--including Korteweg-de Vries, nonlinear Schroedinger, Kuramoto-Sivashinsky, reaction-diffusion, and many others--have a certain structure: the highest spatial derivatives appear only linearly. These high derivatives are a source of stiffness, causing severe time step restrictions for classical numerical methods of high accuracy. A number of methods attempt to exploit the linearity of the stiffness, but all involve tradeoffs. I will present methods which use different time integrators in different parts of Fourier space. These methods have high order of accuracy but only mild time step restrictions, and experimentally they outperform the alternatives.

  • Friday, Jan 28
    Tobin Driscoll, University of Colorado
    "Accurate difference methods for computational electromagnetics"

    Numerical simulation is an indispensable tool for electromagnetic studies in science and engineering. Our needs always outstrip our computational resources, so the availability of efficient methods is vital. I will discuss efficient discretizations in both space and time. In space, the key issue is geometry. I will describe composite finite difference methods that implicitly incorporate jump information at interfaces between materials. In time, the goal is to improve on classical methods by exploiting the nature of lossless propagation. This is accomplished by staggering the unknowns in time. In both space and time, computational examples will be used to support the effectiveness of the approaches.

  • Monday, Jan 31
    John Doyle, Caltech
    "Complexity and robustness: Math and physics meets the Internet"

    A great deal of attention has been given recently to describing features of computer networks in terms such as information, entropy, phase transitions, criticality, fractals, self-similarity, power laws, chaos, and so on. While it is now widely accepted that the commonly assumed Poisson traffic models poorly describe internet traffic, it remains to be seen if these insights will lead to new approaches to network protocol design, which remains largely ad hoc. This talk will take a critical look at these trends, and argue that while it may help to describe network traffic using concepts familiar from information theory and statistical physics, the corresponding tools from these subjects are much less useful. Some new results in generalized coding and phase transitions suggest that novel approaches might be successful, however, and that networks provide great opportunities and challenges for the development and testing of new theory. The implications for future convergent, ubiquitous networking and more general complex systems in engineering and biology will also be briefly discussed.

  • Thursday, Feb 3
    Jingfang Huang , MIT
    "Integral Representation of Lattice Sums"

    Lattice structures appear widely in mathematics, physics, biology and chemistry. Examples include electromagnetic scattering by periodic arrays of obstacles, the evaluation of the lattice energy of crystals, the evaluation of elliptic functions, the analysis of the thermodynamic and structural properties of electrolytes, and the computation of periodic solutions to partial differential equations. In this talk, we present new integral formulas for a variety of lattice sums. The method relies on the existence of plane-wave expansions for the governing potential in free-space and can be used for square, rectangular or skewed lattices. Unlike Ewald summation, which yields approximate integral representations, the formulas described here are exact and lower-dimensional. Simple quadrature rules with modest numbers of nodes yield highly accurate results.
  • Friday, Feb 4
    Jingfang Huang , MIT
    "A Fast Multipole Method for the Yukawa Potential"

    We present a fast multipole method for the Yukawa (screened Coulomb) potential. The multipole expansions use spherical Bessel functions in the radial direction combined with spherical harmonics. We derive a diagonal form for the multipole to local translation operator. Compared with direct summation, the work of evaluating all interactions in a system with N particles is reduced from O(N^2) to O(N). Numerical experiments show a break even point of N=1000 for six digit accuracy.
  • Friday, Feb 11 Semyon Tsynkov, Tel Aviv University and ICASE, NASA Langley
    "Active Shielding and Control of the Environmental Noise"

    We present a mathematical framework for the active control of time-harmonic acoustic disturbances. Unlike many existing methodologies, our approach provides for the exact volumetric cancellation of unwanted noise in a given predetermined region of space while leaving unaffected those components of the total acoustic field that are deemed as friendly. The key finding of the work is that for eliminating the unwanted component of the acoustic field in a given area, one needs to know relatively little; in particular, neither the locations nor the structure nor strength of the external noise sources need to be known. The controls are built based solely on the measurements performed on the perimeter of the region to be shielded; moreover, the controls themselves (i.e., additional sources) are concentrated also only near this perimeter. Perhaps as important, the measured quantities can refer to the total acoustic field rather than its unwanted component only, and the methodology can automatically distinguish between the two.

    We construct a closed form general solution to the aforementioned noise control problem. The apparatus used for deriving this general solution is closely connected to the concepts of generalized potentials and boundary projections of Calderon's type. For a given total wave field, the application of the Calderon's projection allows us to decompose it into the incoming and outgoing components with respect to a particular domain of interest, which may have arbitrary shape. Then, the controls are designed so that to suppress the incoming component for the domain to be shielded or alternatively, the outgoing component for the domain, which is complementary to the one to be shielded. To demonstrate that the new noise control technique is appropriate, we analyze a simple two-dimensional model example that allows full analytical consideration.

    As the time permits, we may also discuss the discrete numerical framework for approaching the active shielding problems, the possibilities of optimizing the solution according to different criteria that would fit different practical requirements, the applicability of the new technique to quasi-stationary problems, and future extensions to the cases of the broad-band spectra of disturbances.


  • Friday, Feb 18
    Divakar Viswanath, University of Chicago
    "Random Fibonacci Sequences and the Number 1.13198824..."

    The Fibonacci sequence where every new term is obtained by adding the previous two is known to everyone. The $n$th term of this sequence increases exponentially with $n$ at the rate $1.61803398...$, the famous Golden Ratio.

    Consider random Fibonacci sequences defined by the random recurrence $t_1=t_2=1$ and $t_n = +/- t_{n-1} +/- t_{n-2}$ for $n>2$, where each $+/-$ is independent and either $+$ or $-$ with probability $1/2$. Does the $n$th term in this random sequence level off because of the subtractions? The answer is no. The magnitude of the $n$th term increases exponentially at the rate $1.13198824...$ with probability $1$. I will explain how finding the number $1.13198824...$ involves first a construction related to continued fractions called Stern-Brocot sequences, and then a fractal measure. Random Fibonacci sequences also serve as an example to introduce some remarkable, long-standing results in the theory of random matrix products.

  • Friday, Feb 25


  • Friday, Mar 10
    Lou Kondic, NJIT
    "Instabilities in the flow of thin liquid films"

    I will present the computational results concerning dynamics of the flow of a thin film down an inclined plane within the framework of lubrication approximation. These results provide detailed picture of the flow instabilities and the resulting pattern formation in this nonlinear system. Our efficient, fully nonlinear 2D simulations allow for analysis of large systems for long times and for a direct comparison will available experimental data. One of the goals of this research is to develop closer connection between nonlinear analysis, scientific computing, and the real-life experimental observations (more details, figures and animations are available at http://m.njit.edu/~kondic/thin_films.html)

  • Friday, Mar 17
    Bernardo Cockburn, University of Minnesota
    "Enhanced accuracy by post-processing for finite element methods for hyperbolic equations"

    We show how to enhance the quality of the approximation provided by general finite element methods for transient hyperbolic equations by means of a simple post-processing. The post-processing is a convolution with a kernel whose support has measure of order one if the meshes are arbitrary; if the meshes are locally translation invariant, the support of the kernel is a cube whose edges are of size of order $\Delta x$ only. For example, when polynomials of degree $k$ are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order $k+1/2$ whereas the post-processed approximation is of order $2k+1$; if the exact solution is in L$^2$ only, case in which no order of convergence is available for the DG method, the postprocessed approximate solution is of order $k+1/2$ in L$^2(\Omega_0)$ where $\Omega_0$ is a domain on which the exact solution is smooth enough. Numerical results displaying the sharpness of the estimates are presented.

  • Friday, Mar 24
    Tim Callahan, Umiversity of Michigan
    "Pattern Formation in Weakly Anisotropic Systems"

    Symmetry is an extremely useful tool in the study of pattern forming systems. Many studies of pattern formation assume some form of symmetry, and once this is done various results from group theory and algebraic geometry can be brought to bear on the problem. As no experiment satisfies these symmetries exactly, it is important to be able to take into account the effect of terms that break the symmetry explicitly. We have developed a tool to determine the most general possible effect that an anisotropic term in one representation of a symmetry group can have on a system which resides in another representation. We present two examples of its application to Benard convection: convection subjected to a linear shear flow and convection subjected to differential rotation.

  • Friday, Mar 31
    Rod Little, University of Michigan
    "The Basics of Bayes"

    The Bayesian approach to statistical inference is enjoying somewhat of a revival; see for example the accessible recent article on this topic in Science*. I briefly outline the Bayesian approach to statistical inference and describe how it differs from classical frequentist inference. I then discuss three weaknesses of the frequentist approach to inference, specifically the lack of exact answers except in restrictive classes of problems, the violation of the likelihood principle, and ambiguities concerning the appropriate choice of reference distribution. Finally, strengths and perceived weaknesses of the Bayesian approach to inference are outlined and discussed. The talk will include a number of simple but I think instructive examples concerning inferences about proportions from a single sample and from two independent samples.

    *Malakoff, D. (1999), "Bayes Offers a 'New' Way to Make Sense of Numbers", Science 286, 1460-1464.

  • Friday, Apr 7
    Michael Greenfield, Ford Scientific Research Lab
    "Reverse Monte Carlo and Transition-State Theory Methods for
    Estimating Small-Molecule Diffusivities in Glassy Polymers"

    The diffusivity of a small molecule through a glassy polymer is difficult to predict from first principles, since the characteristic diffusion time scales ($\mu$s) greatly exceed the time scales accessible through molecular dynamics simulations (ns). We have developed a coarse-grained molecular simulation approach for predicting penetrant diffusivity that circumvents this limitation. First, highly multidimensional transition-state theory (TST) is used to calculate the distribution of penetrant jump rates and accompanying polymer chain motions that underlie diffusive motion. Simulating each jump event involves solving a generalized eigenvalue problem at the transition state and then tracking a gradient-following path to the two associated local minima; the frequencies and potential energies at these points determine the jump rate. Next, reverse Monte Carlo is used to create larger networks of interconnected sorption state positions, such that the sorption state concentration, connectivity, and inter-state jump rates satisfy the distributions obtained from the molecular-level TST simulations. Finally, a continuous-time discrete-space Monte Carlo simulation on this network of states leads to the penetrant diffusivity via the Einstein relation.

    Using TST, we found a broad distribution of jump rates, a moderate distribution of state-to-state connectivity, and a narrow distribution of jump lengths. From kinetic Monte Carlo, we found that changes in mean-squared displacement with respect to time depended strongly on the rate constant distribution and jump connectivity distribution. For low average connectivities, an anomalous regime of sublinear diffusion was observed at small times, with the linear regime reached at long times. For moderate average connectivities, a similar anomalous regime existed when a broad distribution of jump rate constants was employed. Our results suggest that an anomalous diffusion regime exists when a few critical paths are rate limiting, as in percolation models. The penetrant diffusivity for times beyond the anomalous regime was of the correct order of magnitude for small-molecule diffusion in a glassy polymer.

    (*) presently at Ford Motor Company, Scientific Research Lab (*) joint work with Doros Theodorou,University of Patras, GREECE