Gibson Grd floor SR

Can one of the most important Italian Renaissance frescoes reduced in hundreds of thousand fragments by a bombing during the Second World War be re-composed after more than 60 years from its damage? Can we reconstruct the missing parts and can we say something about their original color?

In this talk we would like to exemplify, hopefully effectively by taking advantage of the seduction of art, how mathematics today can be applied in real-life problems which were considered unsolvable only few years ago.

The mechanics of thin elastic or viscous objects has applications in e.g. the buckling of engineering structures, the spinning of polymer fibers, or the crumpling of plates and shells. During the past decade the mathematics, mechanics and physics communities have witnessed an upsurge of interest in those issues. A general question is to how patterns are formed in thin structures. In this talk I consider two illustrative problems: the shapes of an elastic knot, and the stitching patterns laid down by a viscous thread falling on a moving belt. These intriguing phenomena can be understood by using a combination of approaches, ranging from numerical to analytical, and based on exact equations or low-dimensional models.

When modeling biochemical reactions within cells, it is vitally important to take into account the effect of intrinsic noise in the system, due to the small copy numbers of some of the chemical species. Deterministic systems can give vastly different types of behaviour for the same parameter sets of reaction rates as their stochastic analogues, giving us an incorrect view of the bifurcation diagram.

Stochastic Simulation Algorithms (SSAs) exist which draw exact trajectories from the Chemical Master Equation (CME). However, these methods can be very computationally expensive, particularly where there is a separation of time scales of the evolution of some of the chemical species. Some of the species may react many times on a time scale for which others are highly unlikely to react at all. Simulating all of these reactions of the fast species is a waste of computational effort, and many different methods exist for reducing the system to one which only contains the slow variables.

In this talk we will introduce the conditional Gillespie algorithm, a method for sampling directly from the conditional distribution on the fast variables, given a static value for the slow variables. Using this, we will go on to describe the constrained Gillespie approach, which uses simulations of the CG algorithm to estimate the drift and diffusion terms of the effective dynamics of the slow variables.

If there is time at the end, I will briefly describe my work on another project, which involves full sampling of the posterior distributions in various problems in data assimilation using Monte Carlo Markov Chain (MCMC) methods.

Presentations by:

09.30 am Bernhard Langwallner Continuum limits of atomistic energies and new computational models of fracture

09.50 am Yasemin Sengul Well-posedness of dynamics

10.10 am Kostas Koumatos X-interfaces and nonclassical austenite-martensite interfaces

10.30 am Tim Squires Models for breast cancer and heart tissue

In a healthy human brain, cerebrospinal fluid (CSF), a water-like liquid, fills a system of cavities, known as ventricles, inside the brain and also surrounds the brain and spinal cord. Abnormalities in CSF dynamics, such as hydrocephalus, are not uncommon and can be fatal for the patient. We will consider two types of models for the so-called infusion test, during which additional fluid is injected into the CSF space at a constant rate, while measuring the pressure continuously, to get an insight into the CSF dynamics of that patient.

In compartment type models, all fluids are lumped into compartments, whose pressure and volume interactions can be modelled with compliances and resistances, equivalent to electric circuits. Since these models have no spatial variation, thus cannot give information such as stresses in the brain tissue, we also consider a model based on the theory of poroelasticity, but including strain-dependent permeability and arterial blood as a second fluid interacting with the CSF only through the porous elastic solid.

This work demonstrates how we can extract clinically useful patterns

extracted from time series data (speech signals) using nonlinear signal
processing and how to exploit those patterns using robust statistical
machine learning tools, in order to estimate remotely and accurately
average Parkinson's disease symptom severity. 

We present an O(N logN) algorithm for the calculation of the first N coefficients in an expansion of an analytic function in Legendre polynomials. In essence, the algorithm consists of an integration of a suitably weighted function along an ellipse, a task which can be accomplished with Fast Fourier Transform, followed by some post-processing.

Optimization problems with time-periodic parabolic PDE constraints can arise in important chemical engineering applications, e.g., in periodic adsorption processes. I will present a novel direct numerical method for this problem class. The main numerical challenges are the high nonlinearity and high dimensionality of the discretized problem. The method is based on Direct Multiple Shooting and inexact Sequential Quadratic Programming with globalization of convergence based on natural level functions. I will highlight the use of a generalized Richardson iteration with a novel two-grid Newton-Picard preconditioner for the solution of the quadratic subproblems. At the end of the talk I will explain the principle of Simulated Moving Bed processes and conclude with numerical results for optimization of such a process.

Efficient linear and infinitely smooth approximation of functions from equidistant samples is a fascinating problem, at least since Runge showed in 1901 that it is not delivered by the interpolating polynomial.

In 1988, I suggested to substitute linear rational for polynomial interpolation by replacing the denominator 1 with a polynomial depending on the nodes, though not on the interpolated function. Unfortunately the so-obtained interpolant converges merely as the square of the mesh size. In 2007, Floater and Hormann have given for every integer a denominator that yields convergence of that prescribed order.

In the present talk I shall present the corresponding interpolant as well as some of its applications to differentiation, integration and the solution of boundary value problems. This is joint work with Georges Klein and Michael Floater.