Wed, 07 Aug 2013

12:00 - 13:00
Gibson Grd floor SR

### An Initial-Boundary Value Problem for the Fully-Coupled Navier-Stokes/Q-Tensor System

Yuning Liu
(University of Regensburg)
Abstract

We will present in this lecture the global existence of weak solutions and the local existence and uniqueness of strong-in-time solutions for the fully-coupled Navier-Stokes/Q-tensor system on a bounded domain $\O\subset\mathbb{R}^d$ ($d=2,3$) with inhomogenerous Dirichlet and Neumann or mixed boundary conditions. Our result is valid for any physical parameter $\xi$ and we consider the Navier-Stokes equations with a general (but smooth) viscosity coefficient.

Fri, 14 Jun 2013

12:00 - 13:00
Gibson Grd floor SR

### On scale-invariant solutions of the Navier-Stokes equations

Vladimir Sverak
(University of Minnesota)
Abstract

The optimal function spaces for the local-in-time well-posedness theory of the Navier-Stokes equations are closely related to the scaling symmetry of the equations. This might appear to be tied to particular methods used in the proofs, but in this talk we will raise the possibility that the equations are actually ill-posed for finite-energy initial data just at the borderline of some of the most benign scale-invariant spaces. This is related to debates about the adequacy of the Leray-Hopf weak solutions for predicting the time evolution of the system. (Joint work with Hao Jia.)

Fri, 31 May 2013

10:00 - 11:00
Gibson Grd floor SR

### Asymptotic Behavior of Problems in Cylindrical Domains - Lecture 4 of 4

Michel Chipot
(University of Zurich)
Abstract

A mini-lecture series consisting of four 1 hour lectures.

We would like to consider asymptotic behaviour of various problems set in cylinders. Let $\Omega_\ell = (-\ell,\ell)\times (-1,1)$ be the simplest cylinder possible. A good model problem is the following. Consider $u_\ell$ the weak solution to $$\cases{ -\partial_{x_1}^2 u_\ell - \partial_{x_2}^2 u_\ell = f(x_2) \quad \hbox{in } \Omega_\ell, \quad \cr \cr u_\ell = 0 \quad \hbox{ on } \quad \partial \Omega_\ell. \cr}$$ When $\ell \to \infty$ is it trues that the solution converges toward $u_\infty$ the solution of the lower dimensional problem below ? $$\cases{ - \partial_{x_2}^2 u_\infty = f(x_2) \quad \hbox{in }(-1,1), \quad \cr \cr u_\infty = 0 \quad \hbox{ on } \quad \partial (-1,1). \cr}$$ If so in what sense ? With what speed of convergence with respect to $\ell$ ? What happens when $f$ is also allowed to depend on $x_1$ ? What happens if $f$ is periodic in $x_1$, is the solution forced to be periodic at the limit ? What happens for general elliptic operators ? For more general cylinders ? For nonlinear problems ? For variational inequalities ? For systems like the Stokes problem or the system of elasticity ? For general problems ? ... We will try to give an update on all these issues and bridge these questions with anisotropic singular perturbations problems. \smallskip \noindent {\bf Prerequisites} : Elementary knowledge on Sobolev Spaces and weak formulation of elliptic problems.
Fri, 24 May 2013

10:00 - 11:00
Gibson Grd floor SR

### Asymptotic Behavior of Problems in Cylindrical Domains - Lecture 3 of 4

Michel Chipot
(University of Zurich)
Abstract

A mini-lecture series consisting of four 1 hour lectures.

We would like to consider asymptotic behaviour of various problems set in cylinders. Let $\Omega_\ell = (-\ell,\ell)\times (-1,1)$ be the simplest cylinder possible. A good model problem is the following. Consider $u_\ell$ the weak solution to $$\cases{ -\partial_{x_1}^2 u_\ell - \partial_{x_2}^2 u_\ell = f(x_2) \quad \hbox{in } \Omega_\ell, \quad \cr \cr u_\ell = 0 \quad \hbox{ on } \quad \partial \Omega_\ell. \cr}$$ When $\ell \to \infty$ is it trues that the solution converges toward $u_\infty$ the solution of the lower dimensional problem below ? $$\cases{ - \partial_{x_2}^2 u_\infty = f(x_2) \quad \hbox{in }(-1,1), \quad \cr \cr u_\infty = 0 \quad \hbox{ on } \quad \partial (-1,1). \cr}$$ If so in what sense ? With what speed of convergence with respect to $\ell$ ? What happens when $f$ is also allowed to depend on $x_1$ ? What happens if $f$ is periodic in $x_1$, is the solution forced to be periodic at the limit ? What happens for general elliptic operators ? For more general cylinders ? For nonlinear problems ? For variational inequalities ? For systems like the Stokes problem or the system of elasticity ? For general problems ? ... We will try to give an update on all these issues and bridge these questions with anisotropic singular perturbations problems. \smallskip \noindent {\bf Prerequisites} : Elementary knowledge on Sobolev Spaces and weak formulation of elliptic problems.
Fri, 17 May 2013

10:00 - 11:00
Gibson Grd floor SR

### Asymptotic Behavior of Problems in Cylindrical Domains - Lecture 2 of 4

Michel Chipot
(University of Zurich)
Abstract

A mini-lecture series consisting of four 1 hour lectures.

We would like to consider asymptotic behaviour of various problems set in cylinders. Let $\Omega_\ell = (-\ell,\ell)\times (-1,1)$ be the simplest cylinder possible. A good model problem is the following. Consider $u_\ell$ the weak solution to $$\cases{ -\partial_{x_1}^2 u_\ell - \partial_{x_2}^2 u_\ell = f(x_2) \quad \hbox{in } \Omega_\ell, \quad \cr \cr u_\ell = 0 \quad \hbox{ on } \quad \partial \Omega_\ell. \cr}$$ When $\ell \to \infty$ is it trues that the solution converges toward $u_\infty$ the solution of the lower dimensional problem below ? $$\cases{ - \partial_{x_2}^2 u_\infty = f(x_2) \quad \hbox{in }(-1,1), \quad \cr \cr u_\infty = 0 \quad \hbox{ on } \quad \partial (-1,1). \cr}$$ If so in what sense ? With what speed of convergence with respect to $\ell$ ? What happens when $f$ is also allowed to depend on $x_1$ ? What happens if $f$ is periodic in $x_1$, is the solution forced to be periodic at the limit ? What happens for general elliptic operators ? For more general cylinders ? For nonlinear problems ? For variational inequalities ? For systems like the Stokes problem or the system of elasticity ? For general problems ? ... We will try to give an update on all these issues and bridge these questions with anisotropic singular perturbations problems. \smallskip \noindent {\bf Prerequisites} : Elementary knowledge on Sobolev Spaces and weak formulation of elliptic problems.
Fri, 10 May 2013

10:00 - 11:00
Gibson Grd floor SR

### Asymptotic Behavior of Problems in Cylindrical Domains - Lecture 1 of 4

Michel Chipot
(University of Zurich)
Abstract

A mini-lecture series consisting of four 1 hour lectures.

We would like to consider asymptotic behaviour of various problems set in cylinders. Let $\Omega_\ell = (-\ell,\ell)\times (-1,1)$ be the simplest cylinder possible. A good model problem is the following. Consider $u_\ell$ the weak solution to $$\cases{ -\partial_{x_1}^2 u_\ell - \partial_{x_2}^2 u_\ell = f(x_2) \quad \hbox{in } \Omega_\ell, \quad \cr \cr u_\ell = 0 \quad \hbox{ on } \quad \partial \Omega_\ell. \cr}$$ When $\ell \to \infty$ is it trues that the solution converges toward $u_\infty$ the solution of the lower dimensional problem below ? $$\cases{ - \partial_{x_2}^2 u_\infty = f(x_2) \quad \hbox{in }(-1,1), \quad \cr \cr u_\infty = 0 \quad \hbox{ on } \quad \partial (-1,1). \cr}$$ If so in what sense ? With what speed of convergence with respect to $\ell$ ? What happens when $f$ is also allowed to depend on $x_1$ ? What happens if $f$ is periodic in $x_1$, is the solution forced to be periodic at the limit ? What happens for general elliptic operators ? For more general cylinders ? For nonlinear problems ? For variational inequalities ? For systems like the Stokes problem or the system of elasticity ? For general problems ? ... We will try to give an update on all these issues and bridge these questions with anisotropic singular perturbations problems. \smallskip \noindent {\bf Prerequisites} : Elementary knowledge on Sobolev Spaces and weak formulation of elliptic problems.
Thu, 13 Jun 2013

14:00 - 15:00
Gibson Grd floor SR

### Lattice rules in a nutshell

Dr Dirk Nuyens
(KU Leuven)
Abstract
Lattice rules are equal-weight quadrature/cubature rules for the approximation of multivariate integrals which use lattice points as the cubature nodes. The quality of such cubature rules is directly related to the discrepancy between the uniform distribution and the discrete distribution of these points in the unit cube, and so, they are a kind of low-discrepancy sampling points. As low-discrepancy based cubature rules look like Monte Carlo rules, except that they use cleverly chosen deterministic points, they are sometimes called quasi-Monte Carlo rules.

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The talk starts by motivating the usage of Monte Carlo and then quasi-Monte Carlo methods after which some more recent developments are discussed. Topics include: worst-case errors in reproducing kernel Hilbert spaces, weighted spaces and the construction of lattice rules and sequences.

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In the minds of many, quasi-Monte Carlo methods seem to share the bad stanza of the Monte Carlo method: a brute force method of last resort with slow order of convergence, i.e., $O(N^{-1/2})$. This is not so.

While the standard rate of convergence for quasi-Monte Carlo is rather slow, being $O(N^{-1})$, the theory shows that these methods achieve the optimal rate of convergence in many interesting function spaces.

E.g., in function spaces with higher smoothness one can have $O(N^{-\alpha})$, $\alpha &gt; 1$. This will be illustrated by numerical examples.

Thu, 06 Jun 2013

14:00 - 15:00
Gibson Grd floor SR

### Discontinuous Galerkin Methods for Modeling the Coastal Ocean

Professor Clint Dawson
(University of Texas at Austin)
Abstract
The coastal ocean contains a diversity of physical and biological

processes, often occurring at vastly different scales. In this talk,

we will outline some of these processes and their mathematical

description. We will then discuss how finite element methods are used

in coastal ocean modeling and recent research into

improvements to these algorithms. We will also highlight some of the

successes of these methods in simulating complex events, such as

hurricane storm surges. Finally, we will outline several interesting

challenges which are ripe for future research.

Thu, 23 May 2013

14:00 - 15:00
Gibson Grd floor SR

### Compressive Imaging: Stable Sampling Strategies using Shearlets

Professor Gitta Kutyniok
(TU Berlin)
Abstract
In imaging science, efficient acquisition of images by few samples with the possibility to precisely recover the complete image is a topic of significant interest. The area of compressed sensing, which in particular advocates random sampling strategies, has had already a tremendous impact on both theory and applications. The necessary requirement for such techniques to be applicable is the sparsity of the original data within some transform domain. Recovery is then achieved by, for instance, $\ell_1$ minimization. Various applications however do not allow complete freedom in the choice of the samples. Take Magnet Resonance Imaging (MRI) for example, which only provides access to Fourier samples. For this particular application, empirical results still showed superior performance of compressed sensing techniques.

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In this talk, we focus on sparse sampling strategies under the constraint that only Fourier samples can be accessed. Since images -- and in particular images from MRI -- are governed by anisotropic features and shearlets do provide optimally sparse approximations of those, compactly supported shearlet systems will be our choice for the reconstruction procedure. Our sampling strategy then exploits a careful variable density sampling of the Fourier samples with $\ell_1$-analysis based reconstruction using shearlets. Our main result provides success guarantees and shows that this sampling and reconstruction strategy is optimal.

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This is joint work with Wang-Q Lim (Technische Universit\"at Berlin).

Thu, 16 May 2013

14:00 - 15:00
Gibson Grd floor SR

### Numerical Modeling of Vesicles: Inertial Flow and Electric Fields

Dr David Salac
(University at Buffalo)
Abstract

The behavior of lipid vesicles is due to a complex interplay between the mechanics of the vesicle membrane, the surrounding fluids, and any external fields which may be present. In this presentation two aspects of vesicle behavior are explored: vesicles in inertial flows and vesicles exposed to electric fields.

The first half of the talk will present work done investigating the behavior of two-dimensional vesicles in inertial flows. A level set representation of the interface is coupled to a Navier-Stokes projection solver. The standard projection method is modified to take into account not only the volume incompressibility of the fluids but also the surface incompressibility of the vesicle membrane. This surface incompressibility is enforced by using the closest point level set method to calculate the surface tension needed to enforce the surface incompressibility. Results indicate that as inertial effects increase vesicle change from a tumbling behavior to a stable tank-treading configuration. The numerical results bear a striking similarity to rigid particles in inertial flows. Using rigid particles as a guide scaling laws are determined for vesicle behavior in inertial flows.

The second half of the talk will move to immersed interface solvers for three-dimensional vesicles exposed to electric fields. The jump conditions for pressure and fluid velocity will be developed for the three-dimensional Stokes flow or constant density Navier-Stokes equations assuming a piecewise continuous viscosity and an inextensible interface. An immersed interface solver is implemented to determine the fluid and membrane state. Analytic test cases have been developed to examine the convergence of the fluids solver.

Time permitting an immersed interface solver developed to calculate the electric field for a vesicle exposed to an electric field will be discussed. Unlike other multiphase systems, vesicle membranes have a time-varying potential which must be taken into account. This potential is implicitly determined along with the overall electric potential field.

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