| Viktoria A. Averina (University of Minnesota) |
Simple mathematics in biomedical applications |
| Abstract: One will rarely find a job listing from a biomedical company
directly asking for a mathematician. Yet many biomedical
applications
ranging from research to manufacturing require a mathematical
aptitude.
The restrictions imposed by physical, clinical and human
factors call
for mathematical solutions to be as simple as possible. I
will give a
brief background introduction to the applications and
describe several
problems in no mathematical depth whatsoever. |
| Xavier Blanc (Université de Paris VI (Pierre et Marie Curie)) |
Fast rotating Bose-Einstein condensates in asymmetric harmonic trap |
| Abstract: A trapped rotating Bose-Einstein condensate is described by minimizing the
Gross-Pitaevskii energy with an angular momentum term. In the fast
rotating regime, one can restrict the minimization space to the lowest
Landau level (LLL), which is the first eigenspace of the linear part of
the Hamiltonian of the system. In the case of a symmetric harmonic trap,
this framework allows to recover, both analytically and numerically, the
lattice of vortices of experiments. In the case of an asymmetric trap, an
LLL can still be defined, but the behaviour is drastically different: the
condensate has no vortex. Furthermore, contrary to the symmetric case,
convergence of minimizers can be proved, and a limit profile can be
computed.
|
| Olivier Dubois (University of Minnesota) |
Minimizing the energy for efficient linear solvers in reservoir simulation |
| Abstract: In modern petroleum reservoir simulators, a significant computational bottleneck is the solution of large linear systems for the pressure variables. The difficulty comes mainly from the strong heterogeneity of the media: the permeability tensor has jumps of several orders of magnitude and is highly anisotropic in typical applications. We demonstrate the use of energy minimizing basis functions to improve the robustness of multigrid and domain decomposition preconditioners in the context of reservoir simulation. These basis functions have the same advantageous properties of other multiscale approaches. In addition, they can be computed with a purely algebraic procedure, which does not require the construction of a geometric coarse mesh.
This is joint work with Ilya D. Mishev (ExxonMobil Upstream Research) and Ludmil Zikatanov (PennState University). |
| Alexander L. Efros (Naval Research Laboratory) |
Surface effect on the quantum size energy levels in
semiconductor nanocrystals |
| Abstract: We study the effect of the surface on the electron and hole energy level structure in spherical semiconductor nanocrystals within 8 band effective mass approximation. The surface properties are modelled by the General Boundary Conditions that allow us to exclude spurious and wing contributions to the eight band envelope function. The boundary conditions contain a surface parameter that is independent of the energy of the electronic states and should be considered as additional to the set of effective mass parameters describing the bulk semiconductor. We have shown that this parameter: (i) effects strongly the size dependence of the electron and hole quantum size energy levels, (ii) changes the symmetry of the lowest energy levels in the valence band, (iii) leads to the existence of surface localized states with energies within the forbidden gap, (iv) induces the spin-orbit splitting of the conduction band states, and (v) causes additional magnetic moment of the electrons. |
| Peter Hinow (University of Minnesota) |
Predicting the drug release kinetics of matrix tablets |
| Abstract: We develop two mathematical models to predict the release kinetics of a water soluble drug from a polymer/excipient matrix tablet. The first of our models consists of a random walk on a weighted graph, where the vertices of the graph represent particles of drug, excipient and polymer, respectively. The graph itself is the contact graph of a multidisperse random sphere packing. The second model describes the dissolution and the subsequent diffusion of the active drug out of a porous matrix using a system of partial differential equations. The predictions of both models show good qualitative agreement with experimental release curves. The models will provide tools for designing better controlled release devices.
This is joint work with Ami Radunskaya (Pomona College), Boris Baeumer (University of Otago) and Ian Tucker (University of Otago). |
| Claude Le Bris (CERMICS) |
Mathematical and computational challenges in Molecular simulation: an overview |
| Abstract: Molecular simulation is increasingly important in many engineering sciences and life sciences. The field has only been recently explored by mathematical analysts and numerical analysts, leading to several achievements, but also leaving major challenging issues unsolved, both theoretically and computationally. The talk will present the state of the art and will review major mathematical and computational issues of practical importance and theoretical relevance. It will also relate such issues of molecular simulation with issues in materials science. It is mostly based on a recent article coauthored with E. Cances and PL. Lions, and published in Nonlinearity, volume 21, T165-T176, 2008. |
| Frédéric Legoll (École Nationale des Ponts-et-Chaussées) |
Effective dynamics using conditional expectations |
| Abstract: We consider a system described by its position Xt, that
evolves according to the overdamped Langevin equation. At equilibrium,
the statistics of X are given by the Boltzmann-Gibbs measure.
Suppose that we are only interested in some given low-dimensional function
ξ(X) of the complete variable (the so-called reaction coordinate).
The statistics of ξ are completely determined by the free
energy associated to this reaction coordinate. In this work, we try
and design an effective dynamics on ξ, that is a low-dimensional
dynamics which is a good approximation of ξ(Xt). Using
conditional expectations, we build an original dynamics, whose
accuracy is supported by error estimates obtained following an
entropy-based approach. Numerical simulations will illustrate
the
accuracy of the proposed dynamics according to various
criteria.
This is joint work with T. Lelievre (ENPC and INRIA). |
| Richard B. Lehoucq (Sandia National Laboratories) |
Peridynamics: a case study for the role of an applied
mathematician at a national lab |
| Abstract: The purpose of my talk is to introduce peridynamics as a proxy for discussing the role of an applied mathematician at a national lab. The peridynamic balance of linear momentum replaces the local source term of the classical continuum balance law with a nonlocal term. The source term represents internal force interaction, and in peridynamics is represented by an integral operator that sums internal forces separated by a finite distance. This integral operator is not a function of the deformation gradient, allowing for a more general notion of deformation than in the classical theory that is well aligned with the kinematic assumptions of molecular dynamics. I review some of the mathematical results achieved during the last two years. |
| Heinz Siedentop (Ludwig-Maximilians-Universität München) |
The ground state energy of atoms: Functionals of the one-particle-reduced density matrix and their relation to the full Schrödinger equation |
| Abstract: To have an explicit formula for the ground state energy (lowest
spectral point) E(Z) of the Schrödinger operator of (neutral)
atoms of atomic number Z is an elusive goal for Z>1 since it is
a matrix differential operator in 3N dimensions with
2z components.
Shortly after the advent of quantum mechanics efforts were made
to reduce the dimensions to 3 and the components to one. The
first steps were taken by Thomas and Fermi and by Hartree and
Fock. A modern version of this idea is due to Hohenberg and
Kohn (density functional theory) and Gilbert (density matrix
functional theory). The price to pay is to give up the
linearity of the problem.
In this talk I will explain the general idea of density matrix
functional theory and show how a particular type of density
matrix functionals (Müller functional and variants thereof, see
the kick-off meeting of IMA's year on mathematics and
chemistry) can be used to get information on the asymptotic
behavior of E(Z). Among other things, we will show, that the
infimum EM(Z) has the same asymptotic expansion
EM(Z) = a Z7/3 + 1/4 Z2 - c Z
5/3+ o(Z5/3)
as the quantum case.
|
| Stephen Wiggins (University of Bristol) |
Recent advances in the high dimensional Hamiltonian dynamics and geometry of reaction dynamics |
| Abstract: In the early development of applied dynamical systems theory it was hoped that the complexity exhibited by low dimensional nonlinear systems might somehow lead to ways of understanding the complex dynamics of high dimensional systems. Unfortunately, there has not been great progress in this area. The availability of high performance computing resources has led to many computational studies of high dimensional systems. But even under these circumstances, the problem of high dimensionality often forces one to make severe assumptions on the dynamics in order to derive physically relevant quantities from the model, e.g., ergodicity assumptions may be necessary in order to deduce a reaction rate from a computation.We approach the problem of high dimensionality from the other direction. Our interest is in the exact Hamiltonian dynamics of high dimensional systems. As applied dynamical systems theory developed and expanded throughout the 70's and 80's there was much effort in applying global, geometrical concepts and techniques to problems related to the dynamics of molecules. In the early 90's this effort began to die out in the chemistry community because the approach did not appear to apply to problems with more than two degrees-of-freedom. New concepts were required. In the past few years there has been much progress along these lines. We will discuss these recent developments and their application to the understanding of a variety of issues related to the dynamics of molecules. Theoretically, we have constructed a dynamically exact /phase space/ transition state theory, for which we can rigorously construct a "surface of (locally) no return" through which all reacting trajectories must pass. It can also be shown that the flux across the surface we construct is minimal. Central to this construction is a normally hyperbolic invariant manifold (NHIM) whose stable and unstable manifolds enclose the phase space conduits of all reacting trajectories. They enable us to determine the volume of trajectories that can escape from a potential well (the "reactive volume"), which is a central quantity in any reaction rate, and to construct a "dynamical" reaction path. Moreover, we show that the NHIM is the mathematical manifestation of the chemist's notion of the "activated complex".
The application of these ideas to concrete problems relies on the computational realisation of these structures. These can be realized locally through the Poincare-Birkhoff normal form, and then globalised. Recent advances in computational techniques enable one to carry out this procedure for systems with a large number of degrees of freedom. A similar set of techniques can be developed to deal with the corresponding quantum mechanical system. In particular a quantum normal form is used to determine quantum mechanical resonances and reaction rates with high precision. In this talk we describe the theory, applications, and computations that make this possible. We will use HCN isomerization and the Muller-Brown potential to illustrate the ideas and methods and point out a number of areas where more close collaborations between chemists and applied mathematicians could prove fruitful. For example, "rare events" from a dynamical systems point of view are homoclinic and heteroclinic trajectories. Are they related, and do they provide insight, into the "rare events" observed in reaction dynamics? |
| Stephen Wiggins (University of Bristol) |
The dynamical systems approach to Lagrangian transport: fronts and eddies in realistic ocean models |
| Abstract: In this talk I will briefly review the dynamical systems approach to Lagrangian transport, with particular emphasis on recent theoretical and computational results that allow the application to realistic ocean models. I will then apply this approach to a study associated with fronts and eddies in the Mediterranean Sea. First, I will discuss the notion of Lagrangian fronts and eddies and their relation to hyperbolic trajectories and their stable and unstable manifolds. After this, I will show how the Lagrangian description allows for a detailed spatio-temporal description of transport. This barely scratches the surface of possible mathematical and physical analyses for such systems and I will conclude by discussing further directions and problems. |
| Wei Xiong (University of Minnesota) |
Hydrodynamics of particles immersed in a thermally fluctuating, viscous, incompressible fluid |
| Abstract: My current work concerns the hydrodynamics of particles immersed in a thermally fluctuating, viscous, incompressible fluid. The governing equations stipulate conservation of momentum in the fluid, conservation of linear and angular momentum of the particle, and no-slip boundary conditions on the boundary of the particle. Is there existence and uniqueness for the solution? What are the limit theorems when time goes to infinity? These problems not only provide more detailed study of physical Brownian motions but also give a testing ground for the techniques in stochastic partial differential equations.
In this talk, I will focus on particles passively advected by the fluid. I will give existence and uniqueness results for passive particles (point particles as well as finite size particles) immersed in the fluid, and I will give limit theorems when the time goes to infinity. Also, I will give numerical simulations of the motion of the particle in the case of deterministic forcing, and a numerical scheme for the stochastic forcing case. |
| Yongmin Zhang |
American option pricing models and obstacle problems |
| Abstract: We first give a brief overview of American option pricing models and numerical methods. We treat American option models as a special class of obstacle problems. Finite element formulation is introduced together with error analysis of numerical solutions. Some interesting properties about sensitivity of the option price to the payoff function are proved. We also give a criterion for the convergence of numerical free boundaries (optimal exercise boundaries) under mesh refinement. Some future research plans will be discussed.
Bio:
Yongmin Zhang is a risk management consultant at Wells Fargo. Prior to the current position, he was a lead research analyst in Capital Market Research Group of Washington Mutual (now part of J. P. Morgan). His area is in fixed income and mortgage analysis. Before he joined this group, he was an assistant professor at State University of New York where he did research in turbulent flow and American options with more than thirty publications and taught numerous courses in applied mathematics and statistics. Prior to this appointment, he was a research scientist at SUNY Research Foundation. He was a co-principle investigator for various grants from US Department of Energy. He holds his Ph.D. in Applied Mathematics from University of Chicago.
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