Clustering, reordering and random graphs
Abstract
From the point of view of a numerical analyst, I will describe some algorithms for:
- clustering data points based on pairwise similarity,
- reordering a sparse matrix to reduce envelope, two-sum or bandwidth,
- reordering nodes in a range-dependent random graph to reflect the range-dependency,
and point out some connections between seemingly disparate solution techniques. These datamining problems arise across a range of disciplines. I will mention a particularly new and important application from bioinformatics concerning the analysis of gene or protein interaction data.
On the stability and convergence of moving mesh methods for a model convection-diffusion problem
Diagonal scaling of discrete differential forms
Abstract
The use of discrete differential forms in the construction of finite element discretisations of the Sobolev spaces H^s, H(div) and H(curl) is now routinely applied by numerical analysts and engineers alike. However, little attention has been paid to the conditioning of the resulting stiffness matrices, particularly in the case of the non-uniform meshes that arise when adaptive refinement algorithms are used. We study this issue and show that the matrices are generally rather poorly conditioned. Typically, diagonal scaling is applied (often unwittingly) as a preconditioner. However, whereas diagonal scaling removes the effect of the mesh non-uniformity in the case of Sobolev spaces H^s, we show this is not so in the case of the spaces H(curl) and H(div). We trace the reason behind this difference, and give a simple remedy for curing the problem.
tba
Abstract
This Seminar has been cancelled and will now take place in Trinity Term, Week 3, 11 MAY.
An excursion through the world of complex networks guided by matrix theory
Abstract
A brief introduction to the field of complex networks is carried out by means of some examples. Then, we focus on the topics of defining and applying centrality measures to characterise the nodes of complex networks. We combine this approach with methods for detecting communities as well as to identify good expansion properties on graphs. All these concepts are formally defined in the presentation. We introduce the subgraph centrality from a combinatorial point of view and then connect it with the theory of graph spectra. Continuing with this line we introduce some modifications to this measure by considering some known matrix functions, e.g., psi matrix functions, as well as new ones introduced here. Finally, we illustrate some examples of applications in particular the identification of essential proteins in proteomic maps.
Saddle point problems in liquid crystal modelling
Abstract
Saddle-point problems occur frequently in liquid crystal modelling. For example, they arise whenever Lagrange multipliers are used for the pointwise-unit-vector constraints in director modelling, or in both general director and order tensor models when an electric field is present that stems from a constant voltage. Furthermore, in a director model with associated constraints and Lagrange multipliers, together with a coupled electric-field interaction, a particular ''double'' saddle-point structure arises. This talk will focus on a simple example of this type and discuss appropriate numerical solution schemes.
This is joint work with Eugene C. Gartland, Jr., Department of Mathematical Sciences, Kent State University.
Flow and Orientation of Nematic Liquid Crystals Described by the Q-Tensor Model
Abstract
The orientational order of a nematic liquid crystal in a spatially inhomogeneous flow situation is best described by a Q-tensor field because of the defects that inevitably occur. The evolution is determined by two equations. The flow is governed by a generalised Stokes equation in which the divergence of the stress tensor also depends on Q and its time derivative. The evolution of Q is governed by a convection-diffusion type equation that contains terms nonlinear in Q that stem from a Landau-de Gennes potential.
In this talk, I will show how the most general evolution equations can be derived from a dissipation principle. Based on this, I will identify a specific model with three viscosity coefficients that allows the contribution of the orientation to the viscous stress to be cast in the form of a Q-dependent body force. This leads to a convenient time-discretised strategy for solving the flow-orientation problem using two alternating steps. First, the flow field of the Stokes flow is computed for a given orientation field. Second, with the given flow field, one time step of the orientation equation is carried out. The new orientation field is then used to compute a new body force which is again used in the Stokes equation and so forth.
For some simple test applications at low Reynolds numbers, it is found that the non-homogeneous orientation of the nematic liquid crystal leads to non-linear flow effects similar to those known from Newtonian flow at high Reynolds numbers.
14:15
Stochastic geometry and telecommunications modelling
Abstract
Stochastic geometry gradually becomes a necessary theoretical tool to model and analyse modern telecommunication systems, very much the same way the queuing theory revolutionised studying the circuit switched telephony in the last century. The reason for this is that the spatial structure of most contemporary networks plays crucial role in their functioning and thus it has to be properly accounted for when doing their performance evaluation, optimisation or deciding the best evolution scenarios. The talk will present some stochastic geometry models and tools currently used in studying modern telecommunications. We outline specifics of wired, wireless fixed and ad-hoc systems and show how the stochastic geometry modelling helps in their analysis and optimisation.