Department of Mathematics

Applied and Interdisciplinary Mathematics Seminar

Winter 2001

The seminar meets on Friday, 3pm , in 3096 East Hall unless otherwise noted.



Questions? Comments? Please email kurganov@math.lsa.umich.edu
or
petrova@math.lsa.umich.edu .



Friday, January 5
Michele Benzi, Emory University
"Solving Large Systems of Linear Equations on Parallel Computers"

Solving very large systems of linear equations is central to many numerical simulations, and is easily the most time-consuming part of the computation. The most common source of large matrix problems remains the discretization (and linearization) of partial differential equations of elliptic and parabolic type. Other areas where large and sparse linear systems arise frequently include the design and computer analysis of circuits, power system networks, chemical engineering processes, macroeconomics models, queueing systems, and others.
One common approach is to design specialized algorithms that are optimal (or nearly so) for a narrow class of problems. Another approach is to developed general-purpose, purely algebraic methods (and software) that achieve reasonable efficiency on a wide range of problems. In this talk I will limit myself to the second approach.
Many sophisticated techniques have been developed in the last fifty years using tools from linear algebra, graph theory, and approximation theory. Currently, the search for efficient solution algorithms is being driven by the need to solve huge systems (with millions and even billions of unknowns!) on massively parallel computers. In this talk I will review some recently proposed methods for solving large-scale sparse linear systems on parallel machines. The emphasis will be on Krylov subspace methods and algebraic preconditioners based on sparse approximate inverses.

  • Friday, January 12
    James Sneyd, University of Michigan and Massey University, New Zealand
    "Calcium Excitability; the Dynamics of Calcium Homeostasis"

    It is well know how active control of the membrane potential leads to complicated dynamical behaviour, and how cells can thus use their membrane potential as a intercellular signalling mechanism. Such electrical communication underlies all neurophysiological systems. What is not so well-known is the fact that calcium plays a similar role as a messenger, both within single cells and between multiple cells. Thus, the study of the dynamic properties of calcium shows many mathematical similarities with models based on electrical excitability, but the physiological mechanisms are quite distinct.
    In response to agonists such as hormones or neurotransmitters, a wide variety of cell types exhibit oscillations in the concentration of intracellular free calcium ions. In larger cells these oscillations often take the form of periodic intracellular waves, and such waves are commonly seen travelling from cell to cell, to form an intercellular calcium wave. The exact physiological function of these calcium waves and oscillations is not completely clear in any cell type. Nevertheless, they are believed to be one important mechanism whereby cells can control their behaviour, and coordinate with their neighbours.
    Not only is the study of calcium oscillations and waves important from the physiological point of view, the mathematics involved is non-trivial and interesting. Most models of calcium waves are of the reaction-diffusion type, and, as yet, very little is known about the properties of travelling wave solutions in these highly non-linear systems. The existence and importance of calcium buffers, and the effects of intercellular boundaries, add additional complications to what is already a difficult problem.
    I shall give a brief overview of the physiology of calcium waves and oscillations, and then discuss how the modelling work has answered (at least partially) specific physiological questions.

  • Friday, January 19
    Peter Smereka, University of Michigan
    "Coupled Oscillators, Landau Damping, and Bubbly Fluids"

    We examine a class of coupled Hamiltonian systems in which identical nonlinear oscillators are coupled through a mean field. This system has a steady incoherent solution which corresponds to the oscillators being completely out of phase with each other. It shown that if the coupling to the mean field is sufficiently small the system will relax weakly to the steady incoherent solution by a process not unlike Landau damping. It also shown that if the coupling is sufficiently large then the incoherent solution is unstable and the oscillators synchronize. Our system shares similarities to the Vlasov-Poisson equation and to Kuramoto's model. We apply this model to understand sound propagation in bubbly fluids.

  • Friday, January 26
    Joachim Rosenthal, University of Notre Dame
    "Reflections on Shannon's Three Challenges"

    In 1948/1949 Claude Shannon wrote two papers [Sha48,Sha49] which became the foundation of modern information theory. The papers showed that information can be compressed up to the `entropy', that data can be transmitted error free at a rate below the capacity and that there exist provable secure cryptographic systems. These were all fundamental theoretical result. The challenge remained to build practical systems which came close to the theoretical optimal systems predicted by Shannon.
    In this overview talk we will explain how the first two challenges concerning coding theory have resulted in practical solutions which are very close to optimal. Then we explain why the gap between the practical implementation of cryptographic protocols with the theoretical result of Shannon is largest.
    The talk will be tutorial in nature and should be accessible to advanced undergraduate students.

    [Sha48] C.E. Shannon, "A mathematical theory of communication", Bell System Tech. J. 27 (1948), 379--423 and 623--656.
    [Sha49] C.E. Shannon, "Communication theory of secrecy systems", Bell System Tech. J. 28 (1949), 656--715.

  • Friday, February 2
    Eric Vanden Eijnden, New York University
    "Random Perturbation of Partial Differential Equations"

    Wentzell-Freidlin theory of large deviations is applied to partial differential equations perturbed by a (weak) random noise. As illustrations, I shall discuss the problem of switching in Ginzburg-Landau systems with bistable potential, the distribution of the velocity in Burgers equation with random forcing, and the distribution of a scalar field advected by a random velocity field.

  • Friday, February 9
    Olof Runborg, Princeton University
    "Multiphase Computations in Geometrical Optics"

    Ray tracing has long been the numerical method of choice for geometrical optics problems. For several reasons, including efficiency concerns, it has recently been challenged by PDE methods based on the non-linear eikonal and transport equations. For this formulation, however, there is no superposition principle and it cannot accomodate solutions with multiple phases, corresponding to crossing rays. We reformulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite we can recover the exact multiphase solution from an associated system of moment equations, closed by an assumption about the form of the density function in the kinetic equation. The moment equations form a hyperbolic system of conservation laws with source terms. Unlike the eikonal equation, the equations will incorporate a finite superposition principle in the sense that if the maximum number of phases is not exceeded, a sum of solutions is also a solution. We present numerical results of how the equations perform on a variety of problems with constant and variable index of refraction.

  • Friday, February 16
    Jacek Miekisz
    University of Warsaw, visiting University of Michigan
    "Stochastic Models in Evolutionary Game Theory"

    In spatial games, players located on vertices of simple graphs adapt to their neighborhoods by choosing best-reply strategies. We will discuss Nash equilibria of such games and their stability against stochastic perturbations. In particular, we will show that stability of an equilibrium may depend on the number of players. To describe the behavior of systems with many interacting players we will use methods of statistical mechanics.

  • Thursday, February 22
    Dimitri D. Vaynblat, Caltech
    "Shock-induced martensitic phase transitions in solids"

    In our work we have developed a theory for phase transitions in solids under shock-loading. (A typical example of such solid-to-solid phase transition is the well known transformation from graphite to diamond, which occurs under appropriate conditions of stress and temperature; similar phenomena are known to occur in iron and many other materials.) This theory applies, in particular, to a class of experiments which, as a result of solid-to-solid phase transitions, give rise to certain characteristic patterns consisting of two shock-like waves. We have shown that the single assumption that stresses in a phase cannot lie beyond the transition boundaries leads to a complete description of the observed phenomena. Our model is different from others proposed in the literature: it does not make use of kinetic relations and it accounts for the observed wave histories without parameter fitting. In the first part of this talk I will focus on the basic mathematical description of our model and present solutions to the complete set of Riemann problems which could arise as a result of dynamic interactions --- including the basic two-wave structures mentioned above. In the second part of the talk I will utilize our Riemann solver to construct general solutions for the piecewise constant initial value problems usually arising in experiment, and I will specialize our solutions to two widely-studied polymorphic phase changes: the graphite--diamond transition and the $\alpha$--$\epsilon$ transition in iron. I will show that, in presence of well-accepted equations of state for the pure phases, our model leads to close quantitative agreement with a wide range of experimental results. Interestingly, in some cases our theory predicts sequences of events which differ from those generally accepted. In particular, our model predicts a variety of regimes for the iron experiments which were not part of previous interpretations, and it indicates the existence of certain unexpected transformation domains in the graphite systems.

  • Friday, February 23
    Martin Greiner, Duke University, Department of Physics
    "From Synthetic towards Real Fully Developed Turbulence -- Learning from Phenomenology"

    Turbulence is one of those notorious problems in classical and nonlinear physics, which so far has not surrendered to the cohorts of physicists. Over almost a century many tactics have been invented to derive the statistical properties of fully developed turbulence from the fundamental hydrodynamical equations, but with little success. Contrary to these theoretical top-down approaches, a bottom-up approach is presented. The emphasis is put on understanding the data and the relationship between various, apparently different data-motivated models first. From there, more sophisticated phenomenological models are developed, which are expected to bridge the gap between data and fundamental hydrodynamical equations.

  • Friday, March 9
    Min-Jun Lai, University of Georgia
    "Bivariate Splines for Numerical Solutions of Navier-Stokes Equations over Polygonal Domains"

    We first introduce bivariate spline spaces of smoothness r>=0 and degree d>r over triangulations. We will use B-form to express these spline functions and explain the smoothness conditions. Then we present a numerical method to use these spline functions for numerical solutions of PDE's for any desirable smoothness and degree. With these preparation, we will be ready to discuss how to use these spline functions to numerically solve Navier-Stokes equations. We will use the stream function formulation and solve nonlinear fourth order elliptic equations. Some numerical experiments will be demonstrated.

  • Friday, March 16
    Mike Waterman, USC
    "DNA Sequence Assembly"



  • Friday, March 23
    Peter Smereka, University of Michigan
    "Mathematical Modeling of Polycrystalline MgO Thin Films Grown by Ion Beam Assisted Deposition"

    One of the promising approaches to a producing superconducting tape using YBCO is to deposit the YBCO onto a MgO substrate. Typically a MgO substrate will be a polycrystalline film with a (001) fiber texture. However, to obtain a high quality superconducting tape it has been determined that one needs MgO films with good in-plane texture. One successful approach is to use ion beam assisted deposition (IBAD) of MgO. This has resulted films with excellent in-plane texture. This talk will describe mathematical models for this problem and for other problems involving growth of polycrystals. The first part of the talk will describe the computation of polycrystals using narrow band level set methods. Computations of IBAD MgO will then be presented. We still need to explore various models of IBAD MgO and the level set approach is, unfortunately, too slow to compute models of IBAD MgO growth with a large number of crystals. For this reason, a simplified computational strategy has been developed which allows for the approximate solution of the mathematical model for IBAD MgO. The results of the simplified approach show that two of the key parameters controlling the in-plane texture are the seed nucleation density and the out-of-plane texture. The model shows that there is an optimal film thickness, after which the in-plane texture will not improve and the surface roughness increases. Furthermore, the MgO surface appears to tilt toward the ion beam. The predictions of the in-plane texture are in reasonable agreement with experiments as is the optimal film thickness. (This is joint work with G. Russo, X. Li, and D. Srolovitz)

  • Friday, March 30
    Charlie Doering, University of Michigan
    "Vicious asexual drunkards on a line: an alternative view of logistic dynamics"

    The logistic differential equation dC/dt = C - C^2 describes the growth and saturation of a population or concentration C(t) of a chemical species of "A" particles in a reacting system with the reversible coagulation process A + A <--> A + B (with the B-species particle concentration in excess, held constant). This familiar dynamics follows from a "mean field" assumption where each individual particle interacts with the entire population. Moreover, the interaction is "reaction limited" in the sense that the overall evolution is controlled by a small probability that coincident reacting particles do indeed react; the system of particles remains well-mixed and spatially homogeneous in a statistical sense. An alternative formulation of this problem is in the "diffusion-controlled" regime where particles react immediately upon contact and the overall kinetics is limited by transport rather than reaction. A system of diffusing particles (drunkards) interacting via the reversible coagulation reaction A + A <--> A (viciously in the A + A --> A part of the reaction, asexually in the A --> A + A part) can be solved exactly in one spatial dimension (on a line). In this talk we will describe the formulation and solution of this interacting particle system, describe a number of non-mean-field aspects of the resulting dynamics --- including a phase transition in the relaxtion kinetics --- and discuss a physical application of the process.

  • Friday, April 6
    Patrick Nelson, University of Michigan
    "Mathematics and AIDS: What can one tell us about the other"

    Delay differential equation models of HIV-1 infection that account for the intracellular delays are more accurate representations of the biology and change the estimated values of kinetic parameters when compared to models without delays. These results are crucial for the developement and administration of antiviral drugs. We developed and analyzed a set of models that include intracellular delays, combination antiretroviral therapy, and the dynamics of both infected and uninfected T cells. We showed that when the drug efficacy is less then perfect the estimated value for the loss rate of productively infected T cells, $\delta$, is increased as much as 23 \% when delay models are used to fit patient data as compared to the values estimated with a non delay model. We provided a mathematical reasoning for this increased value of $\delta$ that is reflected in the levels of productively infected T cells. I will also provide some general results on the stability of non-linear delay differential equation infection models that can be adapted to study hepatitis C, hepatitis B and tuberculosis.

  • Friday, April 13
    TBA