Thursday, February 28, 2008 at 3:00 PM in HA 335
Paul Atzberger, UC-Santa Barbara
Stochastic Immersed Boundary Methods for Simulation of Microscopic Fluid Structure Systems with Thermal Fluctuations
The immersed boundary method is a numerical approach which has been applied to many macroscopic systems involving a fluid which interacts with flexible elastic structures. For microscopic systems of sufficiently small length scale thermal fluctuations become significant and also must be taken into account. In this talk we shall discuss an extension of the immersed boundary method framework which incorporates thermal fluctuations through appropriate stochastic forcing terms in the fluid equations. This gives a system of stiff SPDE\\\'s for which standard numerical methods perform poorly. We shall discuss a few different approaches by which stochastic calculus can be used to obtain analytic results to help in handling the stiff features of the equations. We will further show how this can be used to formulate numerical methods for the fluid-structure equations both discretized on uniform and multilevel adaptive meshes. To demonstrate the approaches in practice we shall present simulation results the microscopic mechanics of polymers, polymer knots, membrane sheets, and vesicles.Tuesday, March 25, 2008 at 3:00 PM in HA 330
Alexander Kurganov, Tulane University
Numerical Methods for Chemotaxis and Haptotaxis Models
I will present new finite-volume and finite-element methods for a class of chemotaxis models and for a closely related haptotaxis model. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the
chemoattractant concentration. It is well-known that solutions of such systems may develop spiky structures or even blow up in finite time provided the total number of cells exceeds a certain threshold. This makes development of numerical methods for chemotaxis systems an extremely delicate and challenging task.
The first step in the derivation of the finite-volume method is made by adding an equation for the chemoattractant concentration gradient to the original system. The convective part of the resulting system is then of a mixed hyperbolic-elliptic type and therefore straightforward numerical methods for the studied system may be unstable. We design a second-order central-upwind scheme for the extended system of PDEs. The scheme is positivity preserving, which is a very important stability property of the method.
In order to derive a finite-element method -- the interior penalty discontinuous Galerkin method -- the chemotaxis system is reformulated in the form of a convection-diffusion-reaction system with a hyperbolic convective part. This form is suitable for designing discontinuous Galerkin methods. We consider Cartesian grids and prove error estimates for the proposed high-order discontinuous Galerkin methods. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution.
Both methods are applied to a number of two-dimensional problems including the most commonly used Keller-Segel chemotaxis model and its modern extensions as well as to a haptotaxis system modeling tumor invasion into surrounding healthy tissue.Wednesday, April 23, 2008 at 3:00 PM in HA 335
Ilse Ipsen, Department of Mathematics, NCState
Coefficients of Ergodicity: An Introduction
So-called «coefficients of ergodicity» were introduced by Markov in 1906 to
describe the long-term behaviour of inhomogeneous Markov chains.
We interpret these coefficients as bounds on eigenvalues and singular
values of stochastic matrices, and extend them to general complex
matrices.
Monday, April 28, 2008 at 2:00 PM in HA 274
Cristobald de Kerchove, Université Catholique de Louvain-la-Neuve, Belgium
Iterative Filtering for a Dynamical Reputation System
The World Wide Web is making more and more use of interactive ratings collected from various users. They evaluate books, movies, other users, etc. This form of voting requires a sort of filtering in order to bring order among reliable and malicious users. I propose a novel iterative method that assigns a reputation to n + m items: n raters and m objects. Each rater evaluates a subset of objects leading to a n x m rating matrix with a certain sparsity pattern. From this rating matrix we give a nonlinear formula to define the reputation of raters and objects. We also provide an iterative algorithm that superlinearly converges to the unique vector of reputations and this for any rating matrix. In contrast to classical outliers detection, no evaluation is discarded in this method but each one is taken into account with different weights for the reputation of the objects. The complexity of one iteration step is linear in the number of evaluations, making our algorithm efficient for large data set. Experiments show good robustness of the reputation of the objects against cheaters and spammers and good detection properties of cheaters and spammers.
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Seminar Organizers: Pierre Gremaud, Ilse Ipsen, Tim Kelley