applied math and analysis, Spring_2012
https://services.math.duke.edu/mcal?listgroup-1
Applied math and analysis Upcoming Seminarsen-us2024-03-28T18:45:05-04:00https://services.math.duke.edu/mcal2024-01-01T12:00:00-05:002dailyMacroscopic limits of a system of self-propelled particles with phase transition
https://services.math.duke.edu/mcal?abstract-7564
The Vicsek model, describing alignment and self-organisation in large
systems of self-propelled particles, such as fish schools or flocks of
birds, has attracted a lot of attention with respect to its simplicity and
its ability to reproduce complex phenomena. We consider here a
time-continuous version of this model, in the spirit of the one proposed by
P. Degond and S. Motsch, but where the rate of alignment is proportional to
the mean speed of the neighboring particles. In the hydrodynamic limit, this
model undergoes a phase transition phenomenon between a disordered and an
ordered phase, when the local density crosses a threshold value. We present
the two different macroscopic limits we can obtain under and over this
threshold, namely a nonlinear diffusion equation for the density, and a
first-order non-conservative hydrodynamic system of evolution equations for
the local density and orientation. (joint work with Pierre Degond and Jian-Guo Liu).<a href="http://arxiv.org/abs/1109.2404">Amic Frouvelle</a> (University of Crete, Greece)2012-01-16T16:30:00-05:007564applied math and analysisApplied Math And Analysis SeminarSpring, 2012Mon, 16 Jan 2012 16:30:00 ESTMonday, January 16, 2012, 4:30pm119 PhysicsMon, 16 Jan 2012 17:30:00 ESTDiffusion maps for changing data
https://services.math.duke.edu/mcal?abstract-7566
Recently there has been a large class of research that utilizes
nonlinear mappings into low dimensional spaces in order to organize potentially
high dimensional data. Examples include, but are not limited to, locally linear
embedding (LLE), ISOMAP, Hessian LLE, Laplacian eigenmaps, and diffusion maps.
In this talk we will focus on the latter, and in particular consider how to
generalize diffusion maps to the setting in which we are given a data set that
evolves over time or changes depending on some set of parameters. Along with
describing the current theory, various synthetic and real world examples will
be presented to illustrate these ideas in practice.<a href="http://users.math.yale.edu/~mh644/">Matthew Hirn</a> (Yale University)2012-01-23T16:30:00-05:007566applied math and analysisData DialogueApplied Math And Analysis SeminarMonday, January 23, 2012, 4:30pmMon, 23 Jan 2012 16:30:00 ESTSpring, 2012119 PhysicsMon, 23 Jan 2012 17:30:00 EST$l_1$-minimization via a generalized Lagrange multiplier algorithm
https://services.math.duke.edu/mcal?abstract-7562
We consider the basis pursuit problem: find the solution of an underdetermined system $Ax=y$ that minimizes the $l_1$-norm. We formulate a min-max principle (that, as we learned, actually goes back to 1970's) based on a Largange multiplier, and propose an iterative shrinkage-thresholding type algorithm that seems to work quite well. We show that the numerical algorithm converges to the exact solution of the basis pursuit problem. We also discuss its application to array imaging in wave propagation. The analysis is based on ODE techniques, regularization and energy methods. This is a joint work with M. Moscoso, A. Novikov and G. Papanicolaou.<a href="http://math.stanford.edu/~ryzhik/">Lenya Ryzhik</a> (Stanford University)2012-01-30T16:30:00-05:007562applied math and analysisApplied Math And Analysis SeminarMon, 30 Jan 2012 17:30:00 EST119 PhysicsSpring, 2012Monday, January 30, 2012, 4:30pmMon, 30 Jan 2012 16:30:00 ESTConsistent signal reconstruction and the geometry of some random polytopes
https://services.math.duke.edu/mcal?abstract-7666
Consistent reconstruction is a linear programming technique for
reconstructing a signal $x\in\RR^d$ from a set of noisy or quantized
linear measurements. In the setting of random frames combined with
noisy measurements, we prove new mean squared error (MSE) bounds for
consistent reconstruction. In particular, we prove that the MSE for
consistent reconstruction is of the optimal order $1/N^2$ where $N$ is
the number of measurements, and we prove bounds on the associated
dimension dependent constant. For comparison, in the important case
of unit-norm tight frames with linear reconstruction (instead of
consistent reconstruction) the mean squared error only satisfies a
weaker bound of order $1/N$. Our results require a mathematical
analysis of random polytopes generated by affine hyperplanes and of
associated coverage processes on the sphere. This is joint work with
Alex Powell.<a href="http://www.math.vanderbilt.edu/~whitehj3/">Tyler Whitehouse</a> (Vanderbilt University)2012-02-06T16:30:00-05:007666applied math and analysisApplied Math And Analysis SeminarSpring, 2012Monday, February 6, 2012, 4:30pmMon, 06 Feb 2012 16:30:00 ESTMon, 06 Feb 2012 17:30:00 EST119 PhysicsGlobal weak solution for kinetic models of active swimming and passive suspensions
https://services.math.duke.edu/mcal?abstract-7616
We investigate two kinetic models for active suspensions of
rod-like and ellipsoidal particles, and passive suspensions of dumbbell
beads dimmers, which couple a Fokker-Planck equation to the incompressible
Navier-Stokes or Stokes equation. By applying cut-off techniques in the
approximate problems and using compactness argument, we prove the existence
of the global weak solutions with finite (relative) entropy for the two and
three dimensional models. For the second model, we establish a new compact
embedding theorem of weighted spaces which is the key in the compactness
argument. (Joint work with Jian-Guo Liu)Xiuqing Chen (Beijing University of Posts & Telecommunications; Duke Physics)2012-02-13T16:30:00-05:007616applied math and analysisApplied Math And Analysis SeminarMon, 13 Feb 2012 17:30:00 EST119 PhysicsMon, 13 Feb 2012 16:30:00 ESTMonday, February 13, 2012, 4:30pmSpring, 2012Can iterative method converge in a finite number of steps?
https://services.math.duke.edu/mcal?abstract-7614
When iterative methods are used to solve a discretized linear system for
partial differential equations, the key issue is how to make the convergence
fast. For different type of problems convergence mechanism can be quite
different. In this talk, I will present an efficient iterative method, the
fast sweeping method, for a class of nonlinear hyperbolic partial
differential equation, Hamilton-Jacobi equation, which is widely used in
optimal control, geometric optics, geophysics, classical mechanics, image
processing, etc. We show that the fast sweeping method can converge in a
finite number of iterations when monotone upwind scheme,  Gauss-Seidel
iterations with causality enforcement and proper orderings are used. We 
analyze its convergence, which is very different from that for iterative
method for elliptic problems. If time permit I will present a new
formulation to compute effective Hamiltonians for homogenization of a class
of Hamilton-Jacobi equations. Both error estimate and stability analysis
will be shown.<a href="http://www.math.uci.edu/~zhao/homepage/home/home.html">Hongkai Zhao</a> (Dept of Mathematics, Univ. of California-Irvine)2012-03-12T16:30:00-04:007614applied math and analysisApplied Math And Analysis SeminarMon, 12 Mar 2012 16:30:00 EDTMonday, March 12, 2012, 4:30pmSpring, 2012Mon, 12 Mar 2012 17:30:00 EDT119 PhysicsEnsemble sampling methods for equilibrium and non-equilibrium problems
https://services.math.duke.edu/mcal?abstract-7740
This talk will survey my efforts with coworkers to develop and analyze
Monte Carlo sampling algorithms for complex (usually high dimensional)
probability distributions. These sampling problems are typically
difficult because they have multiple high probability regions separated
by low probability regions and/or they are badly scaled in the sense
that there are strong unknown relationships between variables. I'll
begin the talk by discussing a simple modification of the standard
diffusion Monte Carlo algorithm that results in a more efficient and
much more flexible tool for use, for example, in rare event simulation.
If time permits I'll discuss a few other ensemble based sampling tools
designed to directly address energy barriers and scaling issues.<a href="http://math.uchicago.edu/~weare/">Jonathan Weare</a> (University of Chicago, Dept of Mathematics)2012-03-19T16:30:00-04:007740applied math and analysisApplied Math And Analysis Seminar119 PhysicsMon, 19 Mar 2012 17:30:00 EDTSpring, 2012Monday, March 19, 2012, 4:30pmMon, 19 Mar 2012 16:30:00 EDTSettling of a Porous Sphere in Stratified Stokes Flow
https://services.math.duke.edu/mcal?abstract-7603
Marine snow, composed of organic and inorganic aggregates, plays a major
role in marine carbon cycling. Most of these macroscopic particles are
extremely porous, allowing diffusion of salt from the ambient fluid to
affect the density and therefore the settling of these particles. In a first
approximation, these particles can be modeled as spheres. This talk will
present a study of the effect of porosity and salt diffusion in the dynamics
of a sphere settling under gravity in a salt-stratfied fluid analytically
and semi-analytically (depending on the ambient density gradient) in
viscosity dominated regimes. For linear stratification, an explicit solution
for the sphere's position in time is derived. For more general ambient fluid
stratification, the sphere's position can be solved for numerically, under
the asymptotic assumptions about the typical time scales of diffusion and
settling. A parametric study of the settling behaviors and preliminary
comparisons with experiments will be presented.<a href="http://www.unc.edu/~khatri">Shilpa Khatri</a> (University of North Carolina at Chapel Hill)2012-03-26T16:30:00-04:007603applied math and analysisApplied Math And Analysis SeminarMon, 26 Mar 2012 17:30:00 EDT119 PhysicsMon, 26 Mar 2012 16:30:00 EDTMonday, March 26, 2012, 4:30pmSpring, 2012Explicit parametrices for time-dependent Fokker-Planck equations
https://services.math.duke.edu/mcal?abstract-7543
We construct explicit approximate Green's functions of
time-dependent, linear Fokker-Planck equations in terms of Dyson series, Taylor
expansions, and exact commutator formulas. Our method gives
an approximate solution that is accurate to arbitrary order in time
in the short-time limit, and it can be extended to large time by bootstrapping.
I will also present some numerical results showing that our
algorithm works well also for degenerate equations such as those arising in pricing of contingent claims.
This is joint work with Victor Nistor and Wen Cheng.<a href="http://www.math.psu.edu/mazzucat/">Anna Mazzucato</a> (Penn State)2012-04-02T16:30:00-04:007543applied math and analysisApplied Math And Analysis SeminarMon, 02 Apr 2012 17:30:00 EDT119 PhysicsSpring, 2012Monday, April 2, 2012, 4:30pmMon, 02 Apr 2012 16:30:00 EDTCan we Quantify & Exploit Tree-like Intermediate Structure in Complex Networks?
https://services.math.duke.edu/mcal?abstract-7733
Large complex networks naturally represent relationships in a variety of settings, e.g. social interactions, computer/communication networks, and genomic sequences. A significant challenge in analyzing these networks has been understanding the intermediate structure those properties not captured by metrics which are local (e.g. clustering coefficient) or global (e.g. degree distribution). It is often this structure which governs the dynamic evolution of the network and behavior of diffusion-like processes on it. Although there is a large body of empirical evidence suggesting that complex networks are often tree-like at intermediate to large size-scales (e.g. work of Boguna et al in physics, Kleinberg on internet routing, and Chung & Lu on power-law graphs), it remains a challenge to take algorithmic advantage of this structure in data analysis. We discuss several approaches and heuristics for quantifying and elucidating tree-like structure in networks, including various tree-decompositions and Gromov's delta hyperbolicity. These approaches were developed with very different "tree-like" applications in mind, and thus we discuss the strengths and short-comings of each in the context of complex networks and how each might aid in identifying intermediate-scale structure in these graphs.<a href="http://www.ornl.gov/~b7r/">Blair Sullivan</a> (Oak Ridge National Laboratory, Computer Science & Mathematics Division)2012-04-16T16:30:00-04:007733applied math and analysisApplied Math And Analysis SeminarMon, 16 Apr 2012 17:30:00 EDT119 PhysicsSpring, 2012Mon, 16 Apr 2012 16:30:00 EDTMonday, April 16, 2012, 4:30pmDispersion in the Presence of Interfacial Discontinuities
https://services.math.duke.edu/mcal?abstract-7761
This talk will focus on probability questions arising in the geophysical
and biological sciences concerning dispersion in highly heterogeneous
environments, as characterized by abrupt changes (discontinuities)
in the diffusion coefficient. Some specific phenomena observed in
laboratory and field experiments involving breakthrough curves (first
passage times), occupation times, and local times will be addressed. This is
based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique
Thomann, and Brian Wood at Oregon State University.<a href="http://www.math.oregonstate.edu/~waymire/">Edward Waymire</a> (Oregon State University)2012-04-23T16:30:00-04:007761applied math and analysisApplied Math And Analysis SeminarSpring, 2012Monday, April 23, 2012, 4:30pmMon, 23 Apr 2012 16:30:00 EDT119 PhysicsMon, 23 Apr 2012 17:30:00 EDTConcentration compactness for the L^2 critical nonlinear Schrodinger equation
https://services.math.duke.edu/mcal?abstract-7677
The nonlinear Schrodinger equation
<table cellspacing=0 border=0 align=center>
<tr>
<td nowrap align="center">
<a name="ref1.1">
i u<sub>t</sub> + <font face=symbol>D</font> u = <font face=symbol>m</font> |u|<sup>(4/d)</sup>u
</td>
<td nowrap align=center>
<a name="eq0"> <font color=blue>(1)</font>
</td>
</tr>
</table>
is said to be mass critical since the scaling u(t,x)=<font face=symbol>l</font><sup>-d/2</sup>
</td>
<td nowrap align=center>
u(t/<font face=symbol>l</font><sup>2</sup>
</td>
<td nowrap align=center>
, x/<font face=symbol>l</font>) preserves the L<sup>2</sup>
</td>
<td nowrap align=center>
- norm, <font face=symbol>m</font> = <font face=symbol>±</font> 1. In this talk we will discuss the concentration compactness method, which is used to prove global well - posedness and scattering for (1) for all initial data u(0) in L<sup>2</sup>
</td>
<td nowrap align=center>
(R<sup>d</sup>
</td>
<td nowrap align=center>
) when <font face=symbol>m</font> = +1, and for u(0) having L<sup>2</sup>
</td>
<td nowrap align=center>
norm below the ground state when <font face=symbol>m</font> = <font face=symbol>-</font>1. This result is sharp.
<p> As time permits the talk will also discuss the energy - critical problem in R<sup>d</sup> \ <font face=symbol>W</font>,
<table cellspacing=0 border=0 align=center>
<tr>
<td nowrap align="center">
<a name="ref1.2">
i u<sub>t</sub> + <font face=symbol>D</font> u = |u|<sup>4/(d <font face=symbol>-</font> 2)</sup> u
</td>
<td nowrap align=center>
,
u|<sub>Bdry(<font face=symbol>W</font>)</sub> = 0,
<a name="eq1"> <font color=blue>(2)</font>
</td>
</tr>
</table>
where <font face=symbol>W</font> is a compact, convex obstacle.<a href="http://math.berkeley.edu/people/faculty/benjamin-dodson">Benjamin Dodson</a> (Dept of Mathematics, Univ. of California-Berkeley)2012-04-30T16:30:00-04:007677applied math and analysisApplied Math And Analysis SeminarMon, 30 Apr 2012 17:30:00 EDT119 PhysicsSpring, 2012Mon, 30 Apr 2012 16:30:00 EDTMonday, April 30, 2012, 4:30pmSampling of Operators
https://services.math.duke.edu/mcal?abstract-7763
Sampling and reconstruction of functions is a central tool in science. A
key result is given by the classical sampling theorem for bandlimited
functions. We describe the recently developed sampling theory for
operators. We call operators bandlimited if their Kohn-Nirenberg symbols
are band limited. The addresses engineers and mathematicians and should be
accessible for those who have some education in linear algebra and
calculus. The talk reviews sampling of functions and introduces some
terminology from the theory of pseudodifferential operators. We will also
discuss sampling theorems for stochastic operators.<a href="http://math.jacobs-university.de/pfander/">Gotz Pfander</a> (Jacobs University)2012-05-07T16:30:00-04:007763applied math and analysisApplied Math And Analysis Seminar119 PhysicsMon, 07 May 2012 17:30:00 EDTMonday, May 7, 2012, 4:30pmMon, 07 May 2012 16:30:00 EDTSpring, 2012TBA [DELETED]
https://services.math.duke.edu/mcal?abstract-12956
<a href="https://sites.google.com/view/jpark776/home">Jaemin Park</a> (University of Basel)2024-04-23T15:15:00-04:0012956Department of Mathematicsapplied math and analysisDepartment of Mathematics SeminarTue, 23 Apr 2024 15:15:00 EDTTuesday, April 23, 2024, 3:15pmSpring, 2024Physics 119Tue, 23 Apr 2024 16:15:00 EDTdeleted