Prof. Bill Rand, Robert H. Smith School of Business, University of Maryland

The Complex Network of Social Media

The dramatic feature of social media is that it gives everyone a voice; anyone can speak out and express their opinion to a crowd of followers with little or no cost or effort, which creates a loud and potentially overwhelming marketplace of ideas. The good news is that the organizations have more data than ever about what their consumers are saying about their brand. The bad news is that this huge amount of data is difficult to sift through. We will look at developing methods that can help sift through this torrent of data and examine important questions, such as who do users trust to provide them with the information that they want? Which entities have the greatest influence on social media users? Using agent-based modeling, machine learning and network analysis we begin to examine and shed light on these questions and develop a deeper understanding of the complex system of social media.

January 31

3:30 PM
Math 3206 (note time + location)

Joint PDE/KI-Net Seminar

Prof. Razvan Fetecau, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge

Swarm dynamics and equilibria for a nonlocal aggregation model

We consider the aggregation equation ρ_{t} − ∇ • (ρ∇K ∗ ρ) = 0 in R^{n}, where the interaction
potential K models short-range repulsion and long-range attraction. We study a family of
interaction potentials with repulsion given by a Newtonian potential and attraction in the form
of a power law. We show global well-posedness of solutions and investigate analytically and
numerically the equilibria and their global stability. The equilibria have biologically relevant
features, such as finite densities and compact support with sharp boundaries. This is joint work
with Yanghong Huang and Theodore Kolokolnikov.

Dr. Carola-Bibiane Schönlieb, Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge

Noise estimation by PDE-constrained optimisation

A key issue in image denoising is an adequate choice of the correct noise model. In a variational approach this amounts to the choice of the data fidelity and its weighting. Depending on this choice, different results are obtained.

In this talk I will discuss a PDE-constrained optimization approach for the determination of the noise distribution in total variation (TV) image denoising. An optimization problem for the determination of the weights correspondent to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied. Additionally, the differentiability of the solution operator is proved and an optimality system characterizing the optimal solutions of each regularized problem is derived. The optimal parameter values are numerically computed by using a quasi-Newton method, together with semismooth Newton type algorithms for the solution of the TV-subproblems. The talk is furnished with numerical examples computed on simulated data.

Fast multiscale total variation flow with applications.

Images consist of features of varying scales. Thus, multiscale image processing techniques are extremely valuable, especially for medical images. We will discuss multiscale image processing techniques based on variational methods, specifically (BV, L^{2}) decompositions. We will discuss the applications to real time denoising, deblurring, inpainting and image registration.

Prof. Anne Gelb, School of Mathematical and Statistical Sciences, Arizona State University

Numerical Approximation Methods for Non-Uniform Fourier Data

In this talk I discuss the reconstruction of compactly supported piecewise smooth functions from non-uniform samples of their Fourier transform. This problem is relevant in applications such as magnetic resonance imaging (MRI) and synthetic aperture radar (SAR).

Two standard reconstruction techniques, convolutional gridding (the non-uniform FFT) and uniform resampling, are summarized, and some of the difficulties are discussed. It is then demonstrated how spectral reprojection can be used to mollify both the Gibbs phenomenon and the error due to the non-uniform sampling. It is further shown that incorporating prior information, such as the internal edges of the underlying function, can greatly improve the reconstruction quality. Finally, an alternative approach to the problem that uses Fourier frames is proposed.

Prof. Weizhu Bao, Department of Mathematics, Center for Computational Science & Engineering,
National University of Singapore

Ground states and dynamics of the nonlinear Schrodinger/Gross-Pitaevskii equations

In this talk, I begin with a brief derivation of the nonlinear
Schrodinger/Gross-Pitaevskii equations
(NLSE/GPE) from Bose-Einstein condensates (BEC) and/or nonlinear optics.
Then I will present some mathematical results on the existence and
uniqueness as well as non-existence of the ground states of NLSE/GPE
under different external potentials and parameter regimes. Dynamical
properties of NLSE/GPE are then discussed, which include conservation
laws, soliton solutions, well-posedness and/or finite time blowup.
Efficient and accurate numerical methods will be presented for computing
numerically the ground states and dynamics. Extension to NLSE/GPE with
an angular momentum rotation term and/or non-local dipole-dipole
interaction will be presented. Finally, applications to collapse and
explosion of BEC, quantum transport and quantized vortex interaction
will be investigated.

Prof. Alex Mahalov, School of Mathematical & Statistical Sciences, Arizona State University

Stochastic 3D Rotating Navier-Stokes Equations: Averaging, Convergence and Regularity

We consider stochastic three-dimensional rotating Navier-Stokes
equations and prove averaging theorems for stochastic problems in the
case of strong rotation. Regularity results are established by
bootstrapping from global regularity of the limit stochastic equations
and convergence theorems. The energy injected in the system by the noise
is large, the initial condition has large energy, and the regularization
time horizon is long. Regularization is the consequence of precise
mechanisms of relevant three-dimensional nonlinear dynamics. We prove
multi-scale averaging theorems for the stochastic dynamics and describe
its effective covariance operator.

Reference
F. Flandoli and A. Mahalov (2012), Stochastic 3D Rotating Navier-Stokes
equations: averaging, convergence and regularity, Archive for Rational
Mechanics and Analysis, vol. 205, Issue 1, p.195-237.

Arbitrarily high order numerical schemes that converge to entropy measure valued solutions of systems of hyperbolic conservation laws.

We start by arguing through numerical examples as to why entropy measure valued solutions are the appropriate solution concept for systems of conservation laws in several space dimensions. Two classes of numerical schemes are presented that can be shown to converge to entropy measure valued solutions. The first class are finite volume schemes based on entropy conservative fluxes and numerical diffusion operators, using a ENO reconstruction. The second class are space-time shock capturing discontinuous Galerkin (STDG) schemes. The schemes are compared on a set of numerical experiments. The lecture concludes with a discussion of efficient ways to compute measure valued solutions using multi-level monte carlo methods, that were originally developed for uncertainty quantification in conservation laws.

Transmission of Antibiotic-Resistant Bacteria in the Health Care Setting

Antibiotic-resistant bacteria present an enormous challenge in hospitals and other health care settings. Infection rates are high and new strains are constantly emerging. Mortality from nosocomial infections of certain gram-negative strains approaches 60%. Mathematical models consisting of systems of differential equations or computer simulations of agent-based models can be used to study interventions that can lessen transmission and infection rates. In this talk, I will discuss techniques for studying nosocomial infections and look at some results from our recent work.

Discontinuous Galerkin method for hyperbolic equations with singularities

Discontinuous Galerkin (DG) methods are finite element
methods with features from high resolution finite difference
and finite volume methodologies and are suitable for
solving hyperbolic equations with nonsmooth solutions.
In this talk we will first give a survey on DG methods,
then we will describe our recent work on the study of DG
methods for solving hyperbolic equations with singularities
in the initial condition, in the source term, or in the
solutions. The type of singularities include both
discontinuities and δ-functions. Especially for
problems involving δ-singularities, many numerical
techniques rely on modifications with smooth kernels and
hence may severely smear such singularities, leading to
large errors in the approximation. On the other hand,
the DG methods are based on weak formulations and can be
designed directly to solve such problems without
modifications, leading to very accurate results.
We will discuss both error estimates for model linear
equations and applications to nonlinear systems including
the rendez-vous systems and pressureless Euler equations
involving δ-singularities in their solutions.
This is joint work with Qiang Zhang, Yang Yang and
Dongming Wei.

Prof. Yann Brenier, Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique

The relative entropy method for geophysical flows

The "relative entropy method" is well-known in several fields of
mathematical physics, PDEs and probabilities (systems of hyperbolic
conservation laws, systems of particles, kinetic equations...). It typically
leads to the rigorous derivation of some asymptotic models having smooth
enough solutions.

Here, we report on examples of geophysical fluid dynamics (with rotation and
convection effects, at different aspect ratio and time scales), for which
the relative entropy method applies in a non-standard way because the
entropy functional depends on the asymptotic solution itself.

Analysis of Oscillations and Defect Measures in Plasma Physics

We perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma represented by the Navier Stokes Poisson system in 3 − D. We show that as the Debye length goes to zero the velocity field strongly converges towards an incompressible velocity vector field and the density fluctuation weakly converges to zero. In general the limit velocity field cannot be expected to satisfy the incompressible Navier Stokes equation, indeed the presence of high frequency oscillations strongly affects the quadratic nonlinearities and we have to take care of self interacting wave packets. We shall provide a detailed mathematical description of the convergence process by using microlocal defect measures and by developing an explicit correctors analysis.

Prof. Andrea Bertozzi, Department of Mathematics, University of California Los Angeles

Swarming by Nature and by Design

The cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, such phenomena have attracted a renewed interest from the engineering community. This talk will cover a survey of the speaker’s research and related work in this area ranging from aggregation models in nonlinear partial differential equations to control algorithms and robotic testbed experiments. We conclude with a discussion of some interesting problems for the applied mathematics community.

Prof. Laurent Desvillettes, École Normale Supérieure de Cachan

Coupling kinetic and fluid equations in the context of the study of sprays

Sprays are complex flows consisting of
an underlying gas and a large quantity of small liquid droplets. They
appear in many industrial devices (engines, nuclear industry) and natural
phenomena (clouds, lungs). Their modeling, first proposed by Williams in
the 70s, can be performed through the coupling of a kinetic equation of
Vlasov type (for the droplets) and a fluid equation (viscous or inviscid,
compressible or incompressible) for the gas. The mathematical theory for
this coupling mixes the kinetic theory (control of moments as in the
Vlasov-Poisson equation) and the Euler/Navier-Stokes theory (strong local
solutions, weak solutions). We wish to present during the seminar some of
the latest results obtained for sprays.

Relative entropy applied to the study of stability of shocks for conservation laws, and application to asymptotic analysis

The relative entropy method is a powerful tool for the study of
conservation laws. It provides, for example, the weak/strong uniqueness
principle, and has been used in different context for the study of
asymptotic limits. Up to now, the method was restricted to the comparison
to Lipschitz solutions. This is because the method is based on the strong
stability in L^{2} of such solutions. Shocks are known to not be strongly
L^{2} stable. We show, however that their profiles are strongly L^{2}
stable up to a drift. We provide a first application of this stability
result to the study of asymptotic limits.

Prof. Gianluca Crippa, Universitaet Basel, Department Mathematik und Informatik

Ordinary Differential Equations and Singular Integrals

Given a Lipschitz vector field, the classical Cauchy-Lipschitz theory gives existence, uniqueness and regularity of the associated ODE flow. In recent years, much attention has been devoted to extensions of such theory to cases in which the vector field is less regular than Lipschitz, but still belongs to some "weak differentiability classes". In this talk, I will review the main points of an approach involving quantitative estimates for flows of Sobolev vector fields (joint work with Camillo De Lellis) and describe a further extension to a case involving singular integrals of L^1 functions (joint work with Francois Bouchut).