A gentle introduction to Quantum Information Science
Quantum Information science concerns the study of the use of quantum
effects in constructing fast computing devices and designing secure
communicating schemes. Advancement in this area will have significant
impact on activities in business, industry, engineering, and many
branches of sciences. For instance, it will lead to great improvement
in security and efficiency issues on financial transaction, data
storage and transmission, network connection, computing and simulation
of quantum systems. In this talk, a gentle introduction of the subject
will be given. Some current research directions will be discussed. (No
quantum mechanics background is needed to understand the lecture.)
Multiphase and multiscale phenomena in magma dynamics
Magma dynamics is the study of coupled mass, momentum, and energy transfer in partially or completely molten terrestrial materials. The governing equations are used to predict the pattern of crystallization of the Earth from an early magma ocean phase, structures in the deep interior of the Earth, modes of volcanism on the surface, and composition of the Earth's interior. New research in magma dynamics shows that microscopic structures in partially molten rocks deeply influence processes that span over hundreds of kilometers in the Earth's deep interior. This talk will present an overview of the current research in the field of magma dynamics with an emphasis on the analytical solution building and numerical techniques.
Anisotropic Diffusion On Sub-Manifolds With Application to Earth Structure Detection
We show a method for computing extendable independent components of stochastic data sets generated by nonlinear mixing.
In particular, we describe the empirical solution of an inverse problem in electro-magnetic measurements of geological formations.
This method suggests a general tool for solving empirically inverse problems.
Joint work with R. Coifman and A. Haddad, Yale University.
Numerical Methods for Geometric Elliptic Partial Differential Equations
Geometric Partial Differential Equations (PDEs) can be used to describe, manipulate and construct shapes based on intrinsic geometric properties such as curvatures, volumes, and geodesic lengths. These equations arise in classical areas of mathematics (Ricci Flow, Surface Theory) and are useful in modern applications (Image Registration, Computer Animation).
In general these equations are considered too difficult to solve, which is why linearized models or other approximations are commonly used. Progress has recently been made in building solvers for a class of Geometric PDEs. These solvers naturally give better geometric results and, in some cases, are competitive in terms of cost with the simplified models.
In the introductory part of the talk I'll give examples of a few important geometric PDEs which can be solved using a numerical method called monotone finite difference schemes: Monge-Ampere, Convex Envelope, Infinity Laplace, Mean Curvature, and others.
These methods have been implemented for registration of Brain Images. For Surface Registration, the Infinity Laplace equation is used to match surfaces using geodesic lengths [Sapiro]. For Volume Registration, the Monge-Ampere equation is used to minimize distortion of volumes [Tannenbaum-Haker-Haber]. Convergent numerical schemes are important in these applications: bad discretizations lead to artificial singularities in the mappings.
Focusing in on the Monge-Ampere equation, which has seen a lot of numerical work recently, I'll show how naive schemes can work well for smooth solutions, but break down in the singular case. This makes having a convergent scheme even more important. Several groups of researchers have proposed numerical schemes which fail to converge, or converge only in the case of smooth solutions. I'll present a convergent solver which is fast: comparable to solving the Laplace equation a few times.
Small scale gaseous hydrodynamics: modeling and simulation
Small scale gaseous hydrodynamics has recently received significant attention both in connection to practical applications (MEMS/NEMS devices) and due to the scientific challenges it poses.
Of particular interest to us is a number of theoretical challenges arising from the breakdown of the Navier-Stokes description when characteristic flow lengthscales approach the fluid internal scale (in this case the molecular mean free path). In this talk, we will discuss recent progress in modeling and simulation methods for small-scale gaseous flows in regimes where the Navier-Stokes description is expected to fail.
On the flow-physics front, we will discuss solutions to basic flow problems which extend our understanding of transport in gaseous systems beyond the Navier-Stokes limit. Both isothermal gas flow and convective heat transfer will be considered. Finally, we will present a recently developed second-order slip model that extends the applicability of the Navier-Stokes description to lengthscales approaching the mean free path scale. Such models are very desirable since analytical and numerical solutions of the Navier-Stokes description are significantly easier to obtain than solutions of the more general Boltzmann equation.
On the simulation front we will present a class of variance-reduced Monte Carlo particle approaches for solving the Boltzmann equation that can simulate arbitrarily small hydrodynamic signals at fixed cost; this is in sharp constrast to typical Monte Carlo simulation methods, such as the prevalent direct simulation Monte Carlo (DSMC), whose cost grows quadratically as the hydrodynamic signal goes to zero.
Mechanisms of length regulation of flagella in Salmonella
The construction of flagellar motors in motile bacteria such as Salmonella is a carefully regulated genetic
process. Among the structures that are built are the hook and the
filament. The length of the hook is tightly controlled while the
length of filaments is less so. However, if a filament is broken off
it will regrow, while a broken hook will not regrow.
The question that will be addressed in this talk is how Salmonella
detects and regulates the length of these structures. This is related
to the more general question of how physical properties (such as size
or length) can be detected by chemical signals and what those
In this talk, I will present mathematical models for the regulation
of hook and filament length. The model for hook length regulation is based on the hypothesis that the hook length is determined by the rate of secretion of the length regulatory molecule FliK and a cleavage reaction with the gatekeeper molecule FlhB. A stochastic model for this interaction is built and analyzed, showing excellent agreement with hook length data. The model for filament length regulation is based on the hypothesis that the growth of filaments is diffusion limited and is measured by negative feedback involving the regulatory protein FlgM. Thus, the model includes diffusion on a one-dimensional domain with a moving boundary, coupled with a negative feedback chemical network. The model shows excellent qualitative agreement with data, although there are some interesting unresolved issues related to the quantitative results.
A great deal of the statistical literature deals with a single sample
coming from a distribution, univariate or multivariate, and the problem
is to identify the distribution or parts thereof by an array of
estimation and testing procedures. As such, this practice neglects
to bring in information from other sources which could improve the
desired inference. An approach which fuses information from many sources
will be presented and applied in various statistical problems,
including time series forecasting.
Melting of the Earth’s mantle, induced by plate divergence at mid-ocean ridges, is responsible for 80% of volcanic activity of our planet and formation the oceanic crust. Although magma production is fairly well understood, it is less clear how magma, which is produced in a broad region underneath the ridge reaches the surface only in a much narrower accretion zone. I will summarize our efforts to understand magma focusing at mid-ocean ridges and how we can use observation of crustal thickness to test these models. Scaling analyses suggest that magma trajectories are essentially vertical underneath most ridges. Focusing can occur if crystallization of magma as it approaches the surface is fast enough to produce a permeability barrier and a decompaction channel beneath that surface, along which magma travels toward the ridge axis. The barrier follows the temperature field of the mantle, which is controlled by the geometry of the ridge. We model barrier geometry and melt trajectories for several regions along the global mid-ocean ridge system to predict along-strike variations in crustal thickness that can be compared to bathymetric and gravity data.
Development of GPU and Mixed-Architecture Algorithms for Theoretical Chemistry
The highest accuracy in the quantum treatment of small chemical reactions is achieved by time-independent methods involving expansion of the wavefunction in a large, overcomplete basis. The difficulty of the ensuing calculations scales as N3, where N is the size of the basis. The extended renormalized Numerov method offers the most efficient propagation technique. This involves, per sector, solution of a full set of linear equations followed by a matrix multiply. We are exploring use of the emergent MAGMA library to implement this algorithm in a GPU environment.
Similarly, determination of all eigenvalues and eigenvectors of large (N > 2000) matrices is a key computational problem in many aspects of theoretical chemistry. We are exploring the efficiency of new algorithms in which the Householder tridiagonalization (followed by iterative Givens rotations to fully diagonal form) is carried out on the CPU while the updates necessary to construct the matrix of eigenvectors are carried out on the GPU.
Compressive Sensing and the Hard Thresholding Pursuit algorithm
This talk provides an overview of the field of Compressive Sensing, which aims at recovering sparse vectors from a seemingly incomplete set of linear measurements. We shall focus on the recovery process rather than the measurement process. First, we will shortly describe the popular 1-minimization algorithm. In particular, the equivalence of the real and complex settings and the case of pre-Gaussian random measurements will be discussed. We will then introduce an iterative algorithm called Hard Thresholding Pursuit, and we will highlight its advantages, namely simplicity, speed, and theoretical performances.
In this talk, I will discuss recent progress and pending issues in understanding the fluctuations of crystal surfaces.
By focusing on a (1+1)-dimensional setting, I will describe the formulation and predictions of stochastically perturbed dynamical systems for surface line defects. Of particular interest in regard to lab experiments is the probability density of distances (gaps) between defects. I will show some analytical results via (i) linearization of the governing equations, and (ii) application of a ``mean field'' approach along with a decorrelation ansatz. The analytical results will be compared to kinetic Monte Carlo simulations.