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 Events
Jose Faro
University of Vigo, Spain
Tue 1 Jun 2010, 2:00pm
Mathematical Biology Seminar
WMAX 110
Analysis of TcR diversity in CD4+ T cells
WMAX 110
Tue 1 Jun 2010, 2:00pm-3:00pm

Abstract


Note for Attendees

preceded by refreshments - please bring your own mug
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Mon 14 Jun 2010, 9:00am SPECIAL
One Time Event
Graduate Studies - Room 200
PhD defense for Amir Moradifam
Graduate Studies - Room 200
Mon 14 Jun 2010, 9:00am-12:00pm
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Mon 14 Jun 2010, 12:30pm SPECIAL
One Time Event
Graduate Studies - Room 200
PhD defense - Craig Cowan
Graduate Studies - Room 200
Mon 14 Jun 2010, 12:30pm-3:30pm
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Mon 14 Jun 2010, 4:00pm SPECIAL
One Time Event
Graduate Studies - Room 200
PhD defense - Ramon Zarate
Graduate Studies - Room 200
Mon 14 Jun 2010, 4:00pm-7:00pm

Details

We apply self-dual variational calculus to inverse problems, optimal
control problems and homogenization problems in partial differential
equations.
Self-dual variational calculus allows for the variational formulation of
equations which are not necessarily of Euler-Lagrange type. Instead, a
monotonicity condition permits the construction of a so called self-dual
Lagrangian. This Lagrangian then permits the construction of a non-negative
functional whose minimum value is zero, and its minimizer is a solution to
the corresponding equation.
In the case of inverse and optimal control problems, we use the functional
given by the self-dual Lagrangian as a penalization function, which
naturally possesses the ideal qualities for such a role. This allows for
the application of standard variational techniques in a convex setting, as
opposed to working with more complex constrained optimization problems.
This extends work pioneered by Barbu and Kunisch.
In the case of homogenization problems, we develop variational
counterparts to results by dal Maso, Piat, Murat and Tartar with the use of
simpler machinery. In this context self-dual variational calculus permits
one to study the asymptotic properties of the potential functional using
classical Gamma-convergence techniques which are simpler to handle than the
direct techniques required to study the asymptotic properties of the
equation itself. The approach also allows for the seamless handling of
multivalued equations.
The study of such problems leads to the introduction of suitable
topological structures of the spaces of maximal monotone operators and
their corresponding self-dual potentials, which we develop in detail
throughout this thesis.

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Auckland
Wed 16 Jun 2010, 4:00pm
Probability Seminar
WMAX 216
Degenerate random environments
WMAX 216
Wed 16 Jun 2010, 4:00pm-5:00pm

Abstract

In joint work with Tom Salisbury, we study a class of random directed graphs on Z^d that includes site percolation and oriented percolation. Motivated by the study of random walks in these random environments, we focus on those models in two dimensions for which the set C_o (sites reachable from the origin) is infinite almost surely. We describe phase transitions in the geometry of C_o and other relevant clusters in some of the more interesting cases.

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UBC
Mon 21 Jun 2010, 3:00pm
Probability Seminar
WMAX 216
The Super Ornstein-Uhlenbeck Process Interacting with its Center of Mass
WMAX 216
Mon 21 Jun 2010, 3:00pm-4:00pm

Abstract

We construct a  supercritical interacting measure-valued diffusion with
representative particles that  interact with the center of mass by using
the historical stochastic calculus of Perkins to modify a super
Ornstein-Uhlenbeck process.
In doing so we prove continuum analogues of
results of Englander (2010) for binary branching Brownian motion.

On the survival set it is  shown, in the attractive case, that the mass
normalized process converges almost surely in the Vasherstein metric
to the stationary distribution of the Ornstein-Uhlenbeck process, centered
at the limiting value of its center of mass. In the repulsive case it is
shown that it converges in probability, provided the repulsion is not too
strong, by appealing to a result of Englander and Winter (2006).

A version of a result of Tribe (1992) is proven on the extinction set;
that is, as it approaches the extinction time, the normalized process
in both the attractive and repulsive cases converges to a random point a.s.
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Andrea Barth
ETH Zurich
Tue 22 Jun 2010, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
WMAX 216
Multi-Level Monte Carlo Finite Element methods for elliptic PDEs with stochastic coefficients
WMAX 216
Tue 22 Jun 2010, 12:30pm-2:00pm

Abstract

It is a well-known property of Monte Carlo methods that quadrupling the sample size halves the error. In the case of simulations of a stochastic partial differential equations, this implies that the total work is the sample size times the discretization costs of the equation. This leads to a convergence rate which is impractical for many simulations, namely in finance, physics and geosciences. With the Multi-level Monte Carlo method introduced herein, the overall work can be reduced to that of the discretization of the equation, which results in the same convergence rate as for the standard Monte Carlo method. The model problem is an elliptic equation with stochastic coefficients. Multi-Level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. Numerical examples complete the theoretical analysis.
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University of Illinois at Urbana-Champaign
Wed 23 Jun 2010, 3:00pm
Topology and related seminars
WMAX 110
A (mostly) intuitive introduction to Goodwillie's Calculus of Functors
WMAX 110
Wed 23 Jun 2010, 3:00pm-4:00pm

Abstract


Abstract: In this talk, we'll use classic definitions from calculus of real variables to motivate what it means to be a polynomial functor of degree n, the construction of the derivative of a functor and the basics of the `Taylor Series' of a functor as well as do a few simple calculations. 
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Tue 29 Jun 2010, 12:30pm SPECIAL
One Time Event
Leon's Lounge - Graduate Student Center
PhD defense - Alan Lindsay
Leon's Lounge - Graduate Student Center
Tue 29 Jun 2010, 12:30pm-3:30pm

Details


In Applied Mathematics, linear and nonlinear eigenvalue problems arise frequently when characterizing the equilibria of various physical systems. In this thesis, two specific problems are studied, the first of which has its roots in micro engineering and concerns Micro-Electro Mechanical Systems (MEMS). A MEMS device consists of an elastic beam deflecting in the presence of an electric field. Modelling such devices leads to nonlinear eigenvalue problems of second and fourth order whose solution properties are investigated by a variety of asymptotic and numerical techniques.

The second problem studied in this thesis considers the optimal strategy for distributing a fixed quantity of resources in a bounded two dimensional domain so as to minimize the probability of extinction of some species evolving in the domain. Mathematically, this involves the study of an indefinite weight eigenvalue problem on an arbitrary two dimensional domain with homogeneous Neumann boundary conditions, and the optimization of the principal eigenvalue of this problem.

Under the assumption that resources are placed on small patches whose area relative to that of the entire domain is small, the underlying eigenvalue problem is solved explicitly using the method of matched asymptotic expansions and several important qualitative results are established.
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Scott MacLachlan
Mathematics, Tufts
Wed 30 Jun 2010, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
WMAX 216
Fast Solvers for Geodynamic Flows
WMAX 216
Wed 30 Jun 2010, 12:30pm-2:00pm

Abstract

Geodynamic flows, such as the convection within the Earth's mantle, are characterized by the extremely viscous nature of the flow, as well as the dependence of the viscosity on temperature. As such, a PDE-based approach, coupling the (stationary) Stokes Equations for viscous flow with a time-dependent energy equation, offers an accurate mathematical model of these flows. While the theory and practice of solving the Stokes Equations is well-understood in the case of an isoviscous fluid, many open questions remain in the variable-viscosity case that is relevant to mantle convection, where large jumps occur in the fluid viscosity over short spatial scales. I will discuss recent progress on developing efficient parallel solvers for geodynamic flows, using algebraic multigrid methods within block-factorization preconditioners.

Note for Attendees

Note unusual day of the week
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Université Paris-Dauphine
Wed 30 Jun 2010, 3:00pm SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
PIMS WMAX 110
Branched transport problems and elliptic approximation
PIMS WMAX 110
Wed 30 Jun 2010, 3:00pm-4:00pm

Abstract

The branched transport problem is the minimization of a concave functional on vecror measures with prescribed divergence. The only admissible measures are those concentrated on 1-rectifiable sets and the energy is the integral of a power $\theta^\alpha$ of their multiplicity $\theta$. I'll present an approximation by Gamma-convergence, through elliptic functionals defined on more regular functions : the idea is minimizing fucntionals such as $\frac 1 \varepsilon \int  |v|^\alpha + \varepsilon |Dv|^2$ under constraints on the divergence of the $H^1$ function $v$. Obviously the exponents on the $\varepsilon$ and on the power of $|v|$ are to be changed if the result wants to be true. This approximation result recalls those of Modica-Mortola for the perimeter functional, where a double-well potential $W$, minimal on $0$ and $1$, is considered, and the energies $\frac 1 \varepsilon \int  W(v)+ \varepsilon |Dv|^2$ converge to the perimeter of the interface between $\{v=0\}$ and $\{v=0\}$. Here the double-well is replaced with a concave power, so that there is a sort of double-well at $0$ and $\infty$. In ths case as well, the energy at the limit concentrates on a lower dimensional structure. Besides the link with the theory of elliptic approximations,  the interest of this convergence lies in its applications for numerics. Actually, we built (in collaboration with E. Oudet, Chambéry) a quite efficient method, which allows to find reasonable local minima of the limit problem, avoiding the NP complications of the usual combinatorial approaches. The Steiner problem of minimal connection may be approached in this way as well.
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