Ilse Ipsen
Department of Mathematics
North Carolina State University
Raleigh, NC, USA

The Mathematics Behind Google's PageRank

How does Google decide in which order to display web pages as the
result of a search?  A major ingredient in this decision is PageRank,
which is a score associated with every web page.  Increasing the
PageRank of a company's web page has become an important factor in
many marketing strategies.

PageRank only depends on the links among web pages, but not on their
content. Mathematically, PageRank can be viewed as the stationary
distribution of a stochastic matrix whose dimension is now in the
hundreds of billions. Hence the computation of PageRank is often
referred to as the world's largest matrix computation.

We discuss the mathematical problem behind PageRank, its numerical
computation, as well as efficient criteria that can guarantee correct
ranking of the PageRank scores. A round off error analysis
demonstrates the validity of the ranking criteria for matrices with
dimension up to 10^(14), and numerical experiments illustrate that
they can effectively rank the top PageRank scores.

Numerical simulation of clogging and back flushing operation in a filter element in liquid filtration

Re-projection: Mathematical and computational challenges for high order
vortex methods

Vortex methods are numerical schemes for approximating solutions to
the Navier-Stokes equations using a linear combination of moving basis
functions to approximate the vorticity field of a fluid. Typically,
the basis function velocity is determined through a Biot-Savart
integral applied at the basis function centroid. Since vortex methods
are naturally adaptive, they are advantageous in flows dominated by
localized regions of vorticity such as jets, wakes and boundary
layers. While they have been successful in numerous engineering
application, the complexity of understanding grid-free methods make
their analysis a uniquely mathematical endeavor. One recent outcome
of rigorous analysis is an new naturally adaptive high order method
with basis functions that deform as they move according to flow
properties. This new class of methods is very unusual because the
basis functions do not move with the physical flow velocity at the
basis function centroid as is usually specified in vortex methods.
One of the leading edge research problems associated with high
accuracy methods of this type is how to re-project extremely deformed
elements onto a configuration of regular elements to prevent
catastrophic growth of interpolation errors. Recent progress in this
area brings together ideas from radial basis function interpolation,
pre-conditioners, image processing, and partial differential equations.

On coupled flow and two-phase flow problems

Résumé/Abstract:

Lieu/Location: McGill

A pruned basis method for exact wavepacket propagation in scattering  
problems with many bound degrees of freedom

Résumé/Abstract:

Lieu/Location: McGill

Connections Between Several Stochastic Reaction Diffusion Methods for Modeling Biochemical Systems

Résumé/Abstract:

Recently three different, but fundamentally related stochastic reaction diffusion models have been used in modeling biochemical systems at the single-cell scale. Two of these methods create realizations of the stochastic process described by the spatially continuous Smoluchowski equation. One, Green's Function Reaction Dynamics, provides exact realizations of this process, while the other, based on a Brownian Dynamics approach, uses a time discretization. Both methods are spatially continuous. An alternative approach used by several authors is based on the reaction-diffusion master equation (RDME); an extension of the well-known chemical master equation. In the RDME approach space is discretized, while time is kept continuous. We will give an overview of each approach, with emphasis on the behavior of the RDME as the lattice spacing is varied. In particular, we will show that while molecules never react in the continuum limit that the lattice spacing approaches zero, for intermediate lattice spacings the RDME can be thought of as an asymptotic approximation to the Smoluchowski equation. Numerical results demonstrating the accuracy of the RDME in approximating the Smoluchowski equation for biologically relevant parameter regimes will be given.

Lieu/Location: McGill

The spectral element method with applications to non-Newtonian flows

Résumé/Abstract:

The spectral element method (SEM) is a high-order method for solving partial differential equations. This talk will begin by outlining some of the key components and advantages of the SEM before describing how it can be utilised to discretize the equations governing fluid flow. Important considerations include the compatibility of the discrete spaces, the treatment of the continuity equation and the efficient solution of the resulting linear systems of equations. The application of the SEM to some benchmark flow problems in non-Newtonian fluid mechanics will be described and the salient features of the schemes will be highlighted. The benchmark problems considered are the flonal centre manifold. Realization theorems in delay-differential equations ask the question of surjectivity of the centre manifold reduction mapping from classes of delay equations to possible classes of ordinary differential equations on the centre manifold. In particular, for a given class of ordinary differential equations on the centre manifold, one wants to find the class of delay equations with the least number of delays for which the centre manifold mapping is surjective. In this talk, I will discuss several realization theorems for linear and nonlinear systems of delay-differential equations and discuss applications to a few mathematical models.

Lieu/Location: McGill

Jayme De Luca
Department of Physics
UFSCAR, Brazil

Variational structure and delay equation of the electromagnetic two-body problem

We will discuss the state-dependent delay equation of the electromagnetic two-body problem. This is a mixed-type delay equation involving two singular denominators, one in the future light-cone and one in the past light-cone. These denominators spam some nontrivial qualitative dynamics with a stiff timescale and are the main hindrance for a numerical integration as an algebraic-differential neutral-delay equation. The problem has a variational structure that differs from the usual Hamiltonian principle of Galilean physics because it needs future and past stories. We shall discuss a natural use of this variational principle to solve the equations as a boundary-value problem and discuss a variational integrator. Last, the variational structure also includes the same denominators, but in a form easier to regularize than the mixed-type equations of motion.

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Application de l'acquisition parcimonieuse à l'imagerie photo-acoustique

Résumé/Abstract:

Les dernières années ont vu émerger de nouvelles techniques permettant la compression quasi-optimale de signaux moyennant certaines contraintes. Plus récemment, cette compression s'est révélée être utile
dans le cadre non seulement du transfert des données mais de leur acquisition même. Dans ce séminaire nous montrerons que la minimisation d'un problème d'optimisation de type L1 permet d'obtenir avec une grande probabilité une représentation parcimonieuse d'images photo-acoustiques. L'application concrète de cet outil permet de développer des techniques d'acquisition d'images biomédicales à un rythme accru en prenant moins de mesures. Des exemples de mesures concrètes seront présentées

Titre

Title: 

Renormalized mesh-free schemes: convergence analysis and applications

Résumé

Les méthodes sans grille, appelées également méthodes particulaires, ont été développées pour
l’approximation des lois de conservation de la dynamique des fluides et des solides et sont maintenant
utilisées dans de nombreux domaines. Le principal avantage de ces méthodes réside dans leur capacité
à s’appliquer avec succès à des simulations complexes impliquant de larges déformations, comme par
exemple des problèmes d’impacts. Les méthodes avec maillage tels que les éléments finis ne sont pas
adaptées à des problèmes tels que les impacts rapides à cause de la structure même du maillage. La
principale idée des méthodes sans grille est de traiter la partie convective des équations séparément
en la résolvant le long des courbes caractéristiques associées au champs de transport à l’aide de
particules numériques. Cet exposé introduit les méthodes sans grille renormalisées pour les lois de
conservation. Les caractéristiques analytiques et numériques de ces méthodes sont présentées ainsi
que leur application à des multimatériaux solides, avec modèle d’endommagement et à des problèmes
de dynamique des fluides.

Abstract:

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Connecting orbits: the missing link in the theory of bursting shear flows

Résumé/Abstract: 

Shear flows, such as flow in a pipe or along a wall, display the phenomenon of bursting. Turbulent flow profiles develop and decay spontaneously at sufficiently high flow rates, even though linear theory predicts that the flow remains smooth. These turbulent bursts increase the drag, with important consequences in many practical applications. Recently, a new theory of bursting was formulated. This theory is based entirely on concepts of nonlinear dynamics, most importantly on heteroclinic connections. Although the qualitative predictions of this theory agree well with numerical and laboratory experiments, no direct computation of such connecting orbits has been performed for realistic models of shear flow. In this talk I will present some preliminary computations on low-order models, as well as ideas for full-scale computations which might verify the new theory of bursting.

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Saint Venant, Beltrami, Volterra and the elasticity complex in a Sobolev space framework for Lipschitz domains.

Résumé/Abstract: 

The characterization of smooth symmetric second order tensor fields that are strain fields was obtained by Saint Venant (1864, necessary conditions)), Beltrami (1886, sufficiency for simply connected domains) and V. Volterra (1906, multiply connected domains). Their results can be interpreted in geometric language in terms of the elasticity complex.

We study the extension of these results to Sobolev spaces and general Lipschitz domains.

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Highly contrasted elastic multi-materials: multi-scale variational modelings and adapted domain decomposition algorithms

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