Landmine detection is a very slow process, largely due to the huge number of false positives with a metal detector, which for safety reasons must all be removed from the ground. Incorporating an additional sensor type such as ground-penetrating radar (GPR) has been shown to reduce the rate of false positives, speeding the clearance process. So far, these GPR detectors have only used very basic imaging methods (or none at all). We discuss implementing a more advanced method, namely Full-Wave Inversion, highlighting some of the difficulties in this nonlinear and ill-conditioned optimisation based method.
In this talk I will give a summary of the results I've obtained in my PhD so far. These include characterisation of the stability of the diffusion flame under gravity and how gravity affects the propagation of the triple flame.
We study the valuation of storage systems where the availability and spot price of the underlying generated are both subject to stochasticity. In particular, application to the storage of electricity from a wind farm with an attached back-up battery is explored; this system comprises two diffusive-type (stochastic) variables, namely the energy production and the electricity spot price, and two time-like variables, specifically the battery operation and time itself. The solution of the related partial differential equation (PDE) model is approached numerically using an efficient algorithm to treat mixed advection and diffusion problems in four dimensions; a semi-Lagrangian based alternating-direction implicit (SLADI) methodology is implemented. Extensive numerical experimentation confirms the method to be robust and yields highly accurate solutions.
An optimal stopping problem is generally formulated as a problem of stopping an ob-
served stochastic process in order to maximise the expected value of a given reward function.
Applications of optimal stopping theory naturally arise in various fields: examples may be
found in biology, engineering, physics, economics and nance. When the underlying stochas-
tic dynamics is a diffusion process, optimal stopping problems reveal a fundamental link to
partial dierential equations, variational inequalities and free-boundary problems.
In the rst part of this seminar I will outline some of the basic features of optimal stopping
theory with particular attention to the case when the observed stochastic process is a solution
of a stochastic dierential equation. I will sketch sucient conditions for the existence of
an optimal stopping time and describe fundamental properties of the value function. In the
second part I will describe the link between optimal stopping and free boundary problems
by providing heuristic, intuitive arguments. In particular we will see how the free-boundary
can in principle provide important information on the optimal stopping strategy.
The last part of the seminar will be devoted to illustrating an example of optimal detec-
tion. In fact, I will discuss how one may use a Brownian motion to detect a hidden target in
an optimal way. It will be shown that the optimal strategy is entirely characterised through
an integral equation for the free-boundary that may be solved numerically.
In this talk we shall consider torsional wave propagation in a pre-stressed annular cylinder. Hydrostatic pressure is applied to the inner and outer surfaces of an incompressible nonlinear elastic annular cylinder, of circular cross-section, whose constitutive behaviour is governed by a Mooney-Rivlin strain energy function. The pressure difference creates an inhomogeneous deformation field and modifies the inner and outer radii of the annular cylinder. We deduce the effect that this pre-stress, and a given axial stretch, has on the propagation of small-amplitude torsional waves through the medium. We use the theory of small-on-large to determine the linear wave equation that governs incremental torsional waves and then determine their dispersion relation. We stress that the inhomogeneous deformation makes the coefficients of the governing ODE spatially dependant and affects the location of the roots of the dispersion relation. We observe that, if the pressure on the outer surface of the annular cylinder is greater (smaller) than that on the inner, then the cut-on frequencies are spaced further apart (closer) than they would be in the stress-free case. This result could potentially be used to tune the propagation characteristics of the cylinder over a range of frequencies.
Motivated by the problem of pulmonary airway reopening, I shall describe some of the features of bubble propagation into rigid and flexible vessels. Key questions to be addressed will include the role of gravity in the system and the influence of wall elasticity.
Miller:Layered media are present in a wide range of imaging applications such as sedimentary rock formations in geophysical ERT and layers of the skin in microscopic EIT. Layers that are finer than the expected resolution, i.e. finer than the electrode spacing, can be modelled as an anisotropic conductivity distribution defined by a spatially varying 3×3 symmetric tensor. It has been proved that the six distinct components of this tensor cannot be uniquely determined even in the case of infinite precision data. In order to overcome this obstacle, we can reduce the number of parameters to be determined by imposing some constraints based on some a priori information of the anisotropy. In the case of layered media, the anisotropy is uniaxial meaning it has one preferred direction. Mathematically, this results in the conductivity tensor having one eigenvalue corresponding to the eigenvector in the direction normal to the layers and one repeated eigenvalue corresponding to the eigenvectors tangent to the layers. Therefore the conductivity tensor at each point is determined uniquely by the two distinct eigenvalues and the vector normal to the layers at that point. We can further reduce the number of parameters to be reconstructed by assuming the direction of the anisotropy is known. This prior information is often available, for example in geophysical applications the direction of thelayers may be known from seismic data and in biomedical applications the layers may be visible for excised tissue or from another imaging modality such as ultrasound for in vivo reconstruction. This approach reduces the problem to that of determining the two eigenvalues at each point. Crabb: We report on a recent pilot study of dynamic lung electrical impedance tomography (EIT) with healthy male volunteers at the University of Manchester. EIT data at 100 frames per second (fps) were obtained from the subjects during controlled breathing, as well as a magnetic resonance image (MRI) to validate the spatial resolution of the EIT reconstruction. The outer surface of the MR image of the torso and electrode positions obtained using MRI fiducial markers informed the construction of a 3D finite element (FE) model extruded along the caudal-distal axis of the subject. Small changes in the boundary that occur during respiration are accounted for by incorporating the sensitivity with respect to boundary shape into a robust temporal difference reconstruction algorithm. The images obtained from both the EIT and MRI modalities were co- registered using the open source medical imaging software, 3D Slicer. A quantitative comparison of reconstruction quality for different EIT reconstructions is achieved through the calculation of the mutual information with a lung-segmented MR image. EIT reconstructions using a linear shape correction algorithm reduce boundary image artefacts, yield better contrast of the lungs, and have greater mutual information by 10% compared with a standard linear EIT reconstruction.
In the framework of the theory of Continuum Mechanics, exact solutions play a fundamental role for several reasons. They allow to investigate in a direct way the physics of various constitutive models, to understand in depth the qualitative characteristics of the differential equations under investigation, and they provide benchmark solutions of complex problems. The Mathematical method used to determine these solutions is usually called the semi-inverse method. This is essentially a heuristic method that consists in formulating a priori a special ansatz on the geometric and/or kinematical fields of interest, and then introducing this ansatz into the field equations. Luck permitting, these field equations reduce to a simple set of equations and then some special boundary value problems may be solved. Here after a short introduction of the theory, I will illustrate, by some examples, some possible dangers/issue inherent to the use of this method.
There's been lots of work done on the theoretical side of cloaking, where waves are guided around an obstacle to make it disappear, sometimes using meta-materials with inhomogeneous density and shear modulus and more recently using pre-stress in a hyperelastic material. However, work is only just starting to appear on the experimental side of this. After a brief intro to the theory of using pre-stress as a cloaking technique I'll go on to discuss the experimental set up and why we aren't currently seeing the results that would be expected.
Roll waves are periodic wave formations which evolve from natural instabilities to the free surface height of a flow of liquid, mud or granular material down an inclined slope and which propagate downslope through a flow with a phase speed which is everywhere greater than the depth-averaged flow velocity. A unique behaviour of roll waves in granular flows has been found in experiments, where material between successive waves slows to rest but the periodicity is retained. We have found that it is possible to model both the cases of continuous roll waves, which are analogous to those in liquid, and stopping roll waves by using a complex friction coefficient.
Geophysical granular flows, such as avalanches, landslides and pyroclastic flows, consist of polydisperse mixtures of different sized particles. As they move large particle are preferentially transported to the surface and sheared to the flow front, where they experience greater frictional forces and are pushed to the sides. This leads to the formation of "finger-like" structures with coarse-particle-rich regions bounding the flow. We will present a model of such phenomena and also show some small scale experiments.