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Logo Manchester

SIAM Afternoon of Talks, Friday 25th March 2011

This meeting is taking place in Frank Adams Room 1.212, Alan Turing Building, University of Manchester at 2pm and the schedule is the following:

Time Table
1400 - 1405 Information about the Manchester SIAM Student Chapter

1405 - 1430

Geoffrey Dawson (University of Manchester)
Trapping and sorting of bubbles using channel geometry. Abstract (click to view).

This study investigates the motion of air bubbles in liquid-filled rectangular channels with sudden changes in channel geometry, by driving the liquid with constant flow rate. Air bubbles moving through a square pipe filled with silicone oil can get trapped as they relax into a broader channel with a sudden expansion, which is as wide as the channel. Experiments were performed to identify the size of bubbles and range of flow rates for which bubbles can get trapped. In order for a bubble to escape the trap by re-entering the smaller channel, the work of the drag forces resulting from the leaky piston flow has to exceed the change in surface energy required to squeeze into the narrower channel. The experimental results were then compared with a simple model of the corresponding energy balance. This trapping mechanism, which allows some bubbles to pass through a contraction paves way for a simple and effective passive sorting mechanism, based on size of bubble and flow rate. This can be easily implemented in microfluidic devices.

1430 - 1455

Xinan Yang (University of Edinburgh)
Approximate Dynamic Programming with Bézier Curves/Surfaces for Top-percentile traffic routing. Talk (click to view). Abstract (click to view).

Multi-homing is used by Internet Service Provider (ISP) to connect to the Internet via different network providers. This study investigates the optimal routing strategy under multi-homing in the case where network providers charge ISPs according to top-percentile pricing (i.e. based on the $\theta$-th highest volume of traffic shipped). We call this problem the Top-percentile Traffic Routing Problem (TpTRP). Solution approaches based on Stochastic Dynamic Programming require discretization in state space, which introduces a large number of state variables. This is known as the curse of dimensionality. To overcome this we suggested to use Approximate Dynamic Programming (ADP) to construct approximations of the value function in previous work, which works nicely for medium size instances of TpTRP. In this work we keep working on the ADP model, use B{\'e}zier Curves/Surfaces to do the aggregation over time. This modification accelerates the efficiency of parameter training in the solution of the ADP model, which makes the real-sized TpTRP tractable.

1455 - 1520

Henry Shum (University of Oxford)
Hydrodynamic simulations of flagellated bacteria swimming near surfaces. Talk (click to view). Abstract (click to view).

Bacteria are ubiquitous on Earth and perform many vital roles in addition to being responsible for a variety of diseases. Locomotion is beneficial for helping the bacterium explore the environment to find nutrient-rich locations and is also crucial in the formation of large colonies, known as biofilms, on solid surfaces immersed in the fluid. Microorganism biofilms have serious implications for industry and medicine. As such, there is considerable interest in understanding the mechanisms behind biofilm formation and activity. Central to the initiation of biofilm formation is the movement of cells towards the substrate. Many bacteria swim by turning corkscrew-shaped flagella. This can be studied computationally by considering hydrodynamic forces acting on the bacterium as the flagellum rotates. Using a boundary element method to solve the Stokes flow equations, it is found that details of the shape of the cell and flagellum affect both swimming efficiency and attraction of the swimmer towards flat no-slip surfaces. For example, simulations show that relatively small changes in cell elongation or flagellum length could make the difference between an affinity for swimming near surfaces and a repulsion.

1520 - 1545 Coffee break

1545 - 1610

Almut Eisenträger (University of Oxford)
Multi-Fluid Poroelastic Modelling of the Cerebrospinal Fluid Infusion Test. Talk (click to view).  Abstract (click to view).

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. Hydrocephalus is a brain disease characterised by accumulation of CSF in the ventricles, accompanied by deformations of the surrounding brain tissue and an increased intracranial pressure (ICP). A standard test on hydrocephalic patients is the so-called infusion test, where additional fluid is injected into the CSF space to obtain information from the pressure response.

We have extended the theory of poroelasticity by including arterial blood pressure in the formulation and combined this multi-fluid poroelastic theory with a spherically symmetric model of the brain to describe the evolution of pressure during an infusion test. We use a strain dependent permeability together with conservation boundary conditions to link the fluid flow and poroelastic elements. The time dependent governing equations are solved using a finite element formulation on a non uniform mesh. The resulting simulation of ICP during the infusion test is compared to clinical data and further information about strain and fluid content distribution across the brain can be calculated.

1610 - 1635

Waqquas Ahmed Bukhsh (University of Edinburgh)
An MILP Approach to Power Systems Optimization. Talk (click to view).   Abstract (click to view).

Deregulation of power industry has introduced new objectives and challenging optimization questions. The optimization problem is nonlinear and non convex. This talk presents a general framework to pose the optimization problems in power systems as a mixed integer linear programming (MILP) problem. The advantage of this formulation is that the convergence to the global optimum is guaranteed. Numerical results show promising performance and potential for application to general MINLP problems.

1635 - 1700

Lijing Lin (University of Manchester)
Fractional Powers of a Matrix. Talk (click to view).   Abstract (click to view).

talk_lijing_2011 The aim of this work is to devise a reliable algorithm for computing $ A^p$ for $ A\in\mathbb{C}^{n\times n}$ and arbitrary $ p\in\mathbb{R}$. The need to compute fractional powers $ A^p$ arises in a variety of applications, including Markov chain models in finance and healthcare, fractional differential equations, discrete representations of norms corresponding to finite element discretizations of fractional Sobolev spaces, and the computation of geodesic-midpoints in neural networks. Here, $ p$ is an arbitrary real number, not necessarily rational. Often, $ p$ is the reciprocal of a positive integer $ q$, in which case $ X = A^p = A^{1/q}$ is a $ q$th root of $ A$. Various methods are available for the $ q$th root problem. However, none of these methods is applicable for arbitrary real $ p$. In this work, a new algorithm is developed for computing arbitrary real powers $ A^p$. The algorithm starts with a Schur decomposition, takes $ k$ square roots of the triangular factor $ T$, evaluates an $ [m/m]$ Padé approximant of $ (1-x)^p$ at $ I -
T^{1/2^k}$, and squares the result $ k$ times. The parameters $ k$ and $ m$ are chosen to minimize the cost subject to achieving double precision accuracy in the evaluation of the Padé approximant, making use of a result that bounds the error in the matrix Padé approximant by the error in the scalar Padé approximant with argument the norm of the matrix. The Padé approximant is evaluated from the continued fraction representation in bottom-up fashion, which is shown to be numerically stable. In the squaring phase the diagonal and first superdiagonal are computed from explicit formulae for $ T^{p/2^j}$, yielding increased accuracy. Since the basic algorithm is designed for $ p\in(-1,1)$, a criterion for reducing an arbitrary real $ p$ to this range is developed, making use of bounds for the condition number of the $ A^p$ problem. How best to compute $ A^k$ for a negative integer $ k$ is also investigated. In numerical experiments the new algorithm is found to be superior in accuracy and stability to several alternatives, including the use of an eigendecomposition and approaches based on the formula $ A^p =\exp(p\log(A))$.

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