Description

Simple random sampling. Sampling for proportions. Estimation of sample size. Sampling with varying probabilities. Sampling: stratified, systematic, cluster, double, sequential, incomplete. Also listed as SURV 440.

Prerequisites

STAT 401 or STAT 420

Topics

Basic concepts

Populations, samples, sampling frames.
Sampling design, statistics, bias.
Sampling and nonsampling errors.
(0.5 week)

Simple Random Sampling

Estimates of population mean, total, proportion and variance and their sampling properties.
Confidence limits, use of normal approximation.
Auxiliary information, ratio and regression estimators.
(4.5 weeks)

Stratified Samples

Definitions, weighting and estimators.
Optimal allocation, poststratification.
(2 weeks)

Unbiased Estimation for Cluster and Two-Stage Sampling

Single-stage, two-stage, and multi-stage cluster sampling.
Fixed and random clusters.
With-replacement and without-replacement sampling of PSU's.
Approximate variance estimators.
(4 weeks)

Advanced Topics

Variance estimation, categorical data analysis, regression in complex surveys.

  • Homogenization of Integral Energies Under Periodically Oscillating Differential Constraints


    Speaker: Prof. Irene Fonseca (Department of Mathematical Science, Carnegie Mellon University) - http://www.math.cmu.edu/math/faculty/Fonseca

    When: Thu, May 5, 2016 - 3:30pm
    Where: Math 3206

    View Abstract

    Abstract: A homogenization result for a family of integral energies is presented, where the fields are subjected to periodic first order oscillating differential constraints in divergence form. We will give an example that illustrates that, in general, when the operators differential operators have non constant coefficients then the homogenized functional maybe be nonlocal, even when the energy density is convex. This is joint work with Elisa Davoli, and is based on the theory of A-quasiconvexity with variable coefficients and on two-scale convergence techniques.
  • Quantum Dots and Dislocations: Dynamics of Materials Defects


    Speaker: Aziz Lecture: Prof. Irene Fonseca (Department of Mathematical Science, Carnegie Mellon University) - http://www.math.cmu.edu/math/faculty/Fonseca

    When: Wed, May 4, 2016 - 3:15pm
    Where: Math 3206

    View Abstract

    Abstract: The formation and assembly patterns of quantum dots have a significant impact on the optoelectronic properties of semiconductors. We will address short time existence for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained three-dimensional films. Further, the nucleation of misfit dislocations will be analyzed. This is joint work with Nicola Fusco, Giovanni Leoni and Massimiliano Morini.
  • On ill-posedness of Euler system with non-local terms


    Speaker: Agnieszka Swierczewska-Gwiazda (Institute of Applied Mathematics and Mechanics, University of Warsaw (joint KI-Net PDE-Applied Math seminar)) -

    When: Thu, April 21, 2016 - 3:30pm
    Where: Math 3206

    View Abstract

    Abstract: The talk will concern the issue of existence of weak solutions to the Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity appearing in collective behavior patterns.

    We consider several modifications of the Euler system of fluid dynamics including its pressureless variant driven by non-local interaction repulsive-attractive and alignment forces in the space dimension $N=2,3$. These models arise in the study of self-organisation in collective behavior modeling of animals and crowds.

    We adapt the method of convex integration, adapted to the incompressible Euler system by De Lellis and Szekelyhidi, to show the existence of infinitely many global-in-time weak solutions for any bounded initial data. Then we consider the class of dissipative solutions satisfying, in addition, the associated global energy balance (inequality).

    The discussed result is in a certain sense negative result concerning stability of particular solutions. It turns out that the solutions must be sought in a stronger class than that of weak and/or dissipative solutions. We essentially show that there are infinitely many weak solutions for any initial data and that there is a vast class of velocity fields that gives rise to infinitely many admissible (dissipative) weak solutions. We may therefore infer that the class of weak solutions is not convenient for analysing certain qualitative properties such as stability and formation of the flock patterns. However, we also show that the strong solutions are robust in a larger class of all admissible (dissipative) weak solutions leading to the possibility of establishing certain stability results of flock solutions. We establish a weak-strong uniqueness principle for the pressure driven Euler system with non-local interaction terms as well as for the pressureless system with Newtonian interaction.

    The talk is based on the following result: J. A. Carrillo, E. Feireisl, P. Gwiazda, and A. \'Swierczewska-Gwiazda. Weak solutions for Euler systems with non-local interactions, arXiv:1512.03116
  • A Structure Preserving Discretization For Liquid Crystals With External Fields (joint Numerical Analysis and PDE Seminar)


    Speaker: Prof. Shawn Walker (Department of Mathematics and center for computation and technology, Louisiana State University)) - https://www.math.lsu.edu/~walker/

    When: Thu, April 14, 2016 - 3:30pm
    Where: Math 3206

    View Abstract

    Abstract: We present a finite element method (FEM) for computing equilibrium configurations of liquid crystals with variable degree of orientation. The model consists of a Frank-like energy with an additional "s" parameter that allows for line defects with finite energy, but leads to a degenerate elliptic equation for the director field. Our FEM uses a special discrete form of the energy that does not require regularization, and allows us to obtain a stable (gradient flow) scheme for computing minimizers of the energy. We also include external fields to model the so-called "Freedricksz Transition". Simulations in 2-D and 3-D are presented to illustrate the method.
  • Stochastic Homogenization of Reaction-Diffusion Equations in Stationary-Ergodic and Isotropic Media


    Speaker: Jessica Lin (University of Wisconsin, Madison) -

    When: Thu, April 7, 2016 - 3:30pm
    Where: 3206.0

    View Abstract

    Abstract: We consider reaction-diffusion equations with combustion
    nonlinearity in stationary-ergodic and isotropic environments in
    dimensions $d≤3$. We prove the existence of asymptotic, deterministic
    speeds of propagation for solutions with both spark-like and
    front-like initial data in random heterogeneous media. This leads to a
    general stochastic homogenization result which shows that on average,
    the large-scale large-time behavior is governed by a deterministic
    Hamilton-Jacobi equation modeling front propagation. Applications
    include predicting the evolution of forest fires in random isotropic
    environments. This talk is based on joint work with Andrej Zlatos.

  • A Lagrangian Fluctuation-Dissipation Relation for Scalar Turbulence


    Speaker: Theo Drivas (Johns Hopkins) -

    When: Fri, April 1, 2016 - 2:00pm
    Where: 1313.0

    View Abstract

    Abstract: A common approach to calculate the solution of a scalar advection-diffusion
    equation is by a Feynman-Kac representation which averages over stochastic Lagrangian
    trajectories going backward in time to the initial conditions and boundary data. The trajectories
    are obtained by solving SDE's with the advecting velocity as drift and a backward Itō term representing the scalar diffusivity. In this framework, we present an exact formula for scalar dissipation in terms
    of the variance of the scalar values acquired along each random trajectory. As an important
    application, we study the connection between anomalous scalar dissipation in turbulent flows for
    large Reynolds and Péclet numbers and the spontaneous stochasticity of the Lagrangian particle
    trajectories. The latter property corresponds to the Lagrangian trajectories remaining random
    in the limit Re,Pe→∞, when the backward Itō term formally vanishes but the advecting velocity
    field becomes non-Lipschitz. For flows on domains without boundaries (e.g. tori, spheres) and
    for wall-bounded flows with no-flux Neumann conditions for the scalar, we prove that spontaneous
    stochasticity is necessary and sufficient for anomalous scalar dissipation. The fluctuation-dissipation
    relation provides a Lagrangian representation of scalar dissipation also in turbulent flows where
    present experiments suggest that dissipation is tending to zero as Re,Pe→∞. We discuss an
    illustrative example of Rayleigh-Bénard convection with imposed heat-flux at the top and bottom
    plates. Our formula here shows that the scalar dissipation is given by the variance of the local
    time densities of the stochastic particles at the heated boundaries. The ``ultimate regime'' of
    turbulent convection predicted by Kraichnan-Spiegel occurs when the near-wall particle densities
    are mixed to their asymptotic uniform values in a large-scale turnover time. The current observations
    of vanishing scalar dissipation require that fluid particles be trapped at the wall and remain unmixed
    for many, many large-scale turnover times. This talk presents joint work with Gregory Eyink.
  • The H-principle in Fluid Mechanics


    Speaker: Douglis lecture: Laszlo Szekelyhidi (University of Leipzig) -

    When: Thu, March 31, 2016 - 3:30pm
    Where: Math 3206

    View Abstract

    Abstract: It is known since the pioneering work of Scheffer and Shnirelman that weak solutions of the incompressible Euler equations exhibit a wild behaviour, which is very different from that of classical solutions. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to the K41 theory of turbulence.

    In joint work with Camillo De Lellis we established a connection between the theory of weak solutions of the Euler equations and the Nash-Kuiper theorem on rough isometric immersions. Through this connection we interpret the wild behaviour of weak solutions of Euler as an instance of Gromov’s h-principle. In this lecture I will explain this connection and outline recent progress concerning Onsager’s conjecture.
  • Nonlocal reaction-diffusion equations and evolutionary ecology


    Speaker: Olga Turanova (UCLA) -

    When: Thu, March 24, 2016 - 3:30pm
    Where: Math 3206

    View Abstract

    Abstract: We consider a reaction-diffusion equation with a nonlocal reaction term that arises as a model in evolutionary ecology. We study asymptotic behavior and global bounds for solutions of this PDE.

  • Unsteady flow barriers at scales from geophysics to microfluidics: identification, control, and optimizing transport


    Speaker: Sanjeeva Balasuriya (University of Adelaide) -

    When: Thu, February 25, 2016 - 3:30pm
    Where: Math 3206

    View Abstract

    Abstract: Unsteady flows typically possess blobs of particles moving coherently, in addition to regions in which extensive mixing occurs. The boundaries of each of these structures might be considered to be "unsteady flow barriers." Exactly defining what these are is, however, problematic. Such flow barriers may be thought to demarcate geophysical features such as the Antarctic Circumpolar Vortex (ozone hole), or the interface between two fluids that one desires to mix together for DNA synthesis in a microfluidic device. This talk will examine some recent results on the control of unsteady flow barriers, and how one might attempt to optimize mixing across an unsteady flow barrier. Additionally, some ongoing work on how these ideas can be used to correct errors in oceanic velocity data obtained from satellite observations will be briefly discussed.
  • The flow near the ground state for semilinear evolution equations


    Speaker: P. Raphael (University of Nice) -

    When: Tue, February 16, 2016 - 11:00am
    Where: Math 3206

    View Abstract

    Abstract: I will consider the question of the study of the flow near the ground state solitary wave for semilinear heat or Schrodinger type equations. I will illustrate on some recent examples both the problem of construction of non trivial dynamics, in particular singularity formation, and the one of the complete classification of the flow. The approach will allow us to distinguish the construction problem, and in particular the one of minimal elements (with one or possibly more blow up bubbles), and the more involved stability problem. Applications will be given in particular in the mass and energy critical settings.


  • Why gradient flows of some energies good for defect equilibria are not good for dynamics, and an improvement


    Speaker: Prof. Amit Acharya, Joint CSCAMM/Applied Math Seminar (Civil and Environmental Engineering, Carnegie Mellon University) -

    When: Thu, February 11, 2016 - 3:30pm
    Where: Math 3206

    View Abstract

    Abstract: Line defects appear in the microscopic structure of crystalline materials (e.g. metals) as well as liquid crystals, the latter an intermediate phase of matter between liquids and solids. Mathematically, their study is challenging since they correspond to topological singularities that result in blow-up of total energies of finite bodies when utilizing most commonly used classical models of energy density; as a consequence, formulating nonlinear dynamical models (especially pde) for the representation and motion of such defects is a challenge as well. I will discuss the development and implications of a single pde model intended to describe equilibrium states and dynamics of these defects. The model alleviates the nasty singularities mentioned above and it will also be shown that incorporating a conservation law for the topological charge of line defects allows for the correct prediction of some important features of defect dynamics that would not be possible just with the knowledge of an energy function.
  • The Optimal Transportation Problem: applications and numerical methods


    Speaker: Prof. Adam Oberman (McGill University (joint Numerical Analysis/PDE seminar)) - http://www.adamoberman.net/

    When: Thu, February 4, 2016 - 3:30pm
    Where: Math 3206

    View Abstract

    Abstract: The Optimal Transportation problem has been the subject of a great deal of attention by theoreticians in last couple of decades. The Wasserstein (or Earth Mover) distance allows for the metrization of the space of probability measures. However computation of these distances (and the associated maps) has been intractable, except for very small problems.

    Current applications of Optimal Transportation include: Freeform Illumination Optics for shaping light or laser beams, Shape Interpolation (in computer graphics), Machine learning (comparing histograms), discretization of nonlinear PDEs (using the gradient flow in the Wasserstein metric), parameter estimation in geophysics, matching problems in mathematical economics, and Density Functional Theory in physical chemistry.

    Recent advances have allowed for more efficient computation of solutions of the Monge-Kantorovich problem of optimal transportation. In the special, but important case of quadratic costs, the map can be obtained from the solution of the elliptic Monge-Ampere partial differential equation with nonstandard boundary conditions. For more general costs, the Kantorovich plan can be approximated by a finite dimensional linear program. In this talk we will compare the cost and quality of the solutions obtained by two different methods.

    I will also discuss some nonlinear PDE problems (curvature flows, 2-Hessian equation) which can be solved using similar techniques to those applied to the Monge-Ampere PDE.
  • Anomalous diffusion for some kinetic equations


    Speaker: Marjolaine Puel (University of Nice Sophia-Antipolis) -

    When: Thu, November 19, 2015 - 3:30pm
    Where: Math 3206

    View Abstract

    Abstract: Kinetic equations involve a large number of variables, time, space and velocity and one important part of the study of those equation consists in giving an approximation of their solution for large time and large observation length. For example, when we model collisions via the linear Boltzmann equation, it is well known that when the equilibria are given by Gaussian distributions, we can approximate the solution by the product of an equilibrium that gives the dependence with respect to velocity multiplied by a density depending on time and position that satisfies a diffusion equation. But different models like inelastic collisions lead to heavy tails equilibria for which depending on the power of the tail, we get different situations. When the diffusion coefficient is no more defined, in the case of linear Boltzmann, the density satisfies a fractional diffusion equation. The same kind of problem arises when the interaction between particles are modeled via the Fokker Planck operator with an additional difficulty. I will present a probabilistic method to study the critical case where we obtain still a diffusion but with an anomalous scaling and present the problems arising for the subcritical exponents.

  • Energy scaling laws for compressed thin elastic sheets


    Speaker: Ian Tobasco (Courant Institute) -

    When: Thu, November 5, 2015 - 3:30pm
    Where: Math 3206

    View Abstract

    Abstract: A long-standing open problem in elasticity is to identify the minimum energy scaling law of a crumpled sheet of paper as thickness tends to zero. Though much is known about scaling laws for thin sheets in tensile settings, the compressive regime is mostly unexplored. I will discuss the analysis of two examples: an axially compressed thin elastic cylinder, and an indented cone. My focus in this talk will be the dependence of the minimum energy on the thickness and loading in the Foppl-von Karman model. I will prove upper and lower bounds for these scalings. The material for this talk is drawn from two papers in preparation; the work on indented cones is in collaboration with H. Olbermann and S. Conti.
  • Spectral representation of generalized Laguerre semigroups and hypocoercitivity


    Speaker: Pierre Patie (Cornell University) -

    When: Thu, October 1, 2015 - 3:30pm
    Where: Math 3206

    View Abstract

    Abstract: The first aim of this to talk is to present an original methodology for developing the spectral representation of a class of non-self-adjoint (NSA) invariant semigroups. This class is defined in terms of self-similar semigroups on the positive real line and we name it the class of generalized Laguerre semigroups. Our approach is based on an in-depth analysis of an intertwinning relationship that we establish between this class and the classical Laguerre semigroup which is self-adjoint. We proceed by discussing substantial difficulties that one may face when studying the spectral representation of NSA operators.
    Finally, we also show that our approach enables us to get precise information regarding the speed of convergence towards stationarity. In particular, we observe in some cases the hypocoercivity phenomena which, in our context, can be interpreted in terms of the spectral norms.

  • Transonic problems on multidimensional conservation laws


    Speaker: Eun Heui Kim (California State University Long Beach) -

    When: Thu, September 24, 2015 - 3:30pm
    Where: Math 3206
  • Fractional thin film equations and hydraulic fractures


    Speaker: Antoine Mellet (UMD) -

    When: Thu, September 17, 2015 - 3:30pm
    Where: Math 3206
  • Nonlinear Schrödinger equations: The interplay of modeling, numerics and physics


    Speaker: Norbert Mauser (Wolfang Pauli Institute and Univ. of Vienna) -

    When: Fri, August 14, 2015 - 11:00am
    Where: Math 1311