Past Junior Applied Mathematics Seminar

We investigate the behaviour of free-surface waves on time-varying potential flow in the limit as the Froude number becomes small. These waves are exponentially small in the Froude number, and are therefore inaccessible to ordinary asymptotic methods. As such, we demonstrate how exponential asymptotic techniques may be applied to the complexified free surface in order to extract information about the wave behaviour on the free surface, using a Lagrangian form of the potential flow equations. We consider the specific case of time-varying flow over a step, and demonstrate that the results are consistent with the steady state case.

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 explore two different threading approaches on a graphics processing
unit (GPU) exploiting two different characteristics of the current GPU
architecture. The fat thread approach tries to minimise data access time
by relying on shared memory and registers potentially sacrificing
parallelism. The thin thread approach maximises parallelism and tries to
hide access latencies. We apply these two approaches to the parallel
stochastic simulation of chemical reaction systems using the stochastic
simulation algorithm (SSA) by Gillespie. In these cases, the proposed
thin thread approach shows comparable performance while eliminating the
limitation of the reaction system's size.

Link to paper: 

http://people.maths.ox.ac.uk/erban/papers/paperCUDA.pdf

Following work done by the 'Oxford Spies' we uncover more secrets of 'surface-active Agents'. In modern-day applications we refer to these agents as surfactants, which are now extensively used in industrial, chemical, biological and domestic applications. Our work focuses on the dynamic behaviour of surfactant and polymer-surfactant mixtures.

In this talk we propose a mathematical model that incorporates the effects of diffusion, advection and reactions to describe the dynamic behaviour of such systems and apply the model to the over-flowing-cylinder experiment (OFC). We solve the governing equations of the model numerically and, by exploiting large parameters in the model, obtain analytical asymptotic solutions for the concentrations of the bulk species in the system. Thus, these solutions uncover secrets of the 'surface-active Agents' and provide an important insight into the system behaviour, predicting the regimes under which we observe phase transitions of the species in the system. Finally, we suggest how our models can be extended to uncover the secrets of more complex systems in the field.

Colorectal cancer (CRC) is one of the leading causes of cancer-related death worldwide, demanding a response from scientists and clinicians to understand its aetiology and develop effective treatment. CRC is thought to originate via genetic alterations that cause disruption to the cellular dynamics of the crypts of Lieberkűhn, test-tube shaped glands located in both the small and large intestine, which are lined with a monolayer of epithelial cells. It is believed that during colorectal carcinogenesis, dysplastic crypts accumulate mutations that destabilise cell-cell contacts, resulting in crypt buckling and fission. Once weakened, the corrupted structure allows mutated cells to migrate to neighbouring crypts, to break through to the underlying tissue and so aid the growth and malignancy of a tumour. To provide further insight into the tissue-level effects of these genetic mutations, a multi-scale model of the crypt with a realistic, deformable geometry is required. This talk concerns the progress and development of such a model, and its usefulness as a predictive tool to further the understanding of interactions across spatial scales within the context of colorectal cancer.

Graphics processing units (GPU) are well suited to decrease the

computational in-

tensity of stochastic simulation of chemical reaction systems. We

compare Gillespie’s

Direct Method and Gibson-Bruck’s Next Reaction Method on GPUs. The gain

of the

GPU implementation of these algorithms is approximately 120 times faster

than on a

CPU. Furthermore our implementation is integrated into the Systems

Biology Toolbox

for Matlab and acts as a direct replacement of its Matlab based

implementation.

The visceral endoderm (VE) is an epithelium of approximately 200 cells

encompassing the early post-implantation mouse embryo. At embryonic day

5.5, a subset of around 20 cells differentiate into morphologically

distinct tissue, known as the anterior visceral endoderm (AVE), and

migrate away from the distal tip, stopping abruptly at the future

anterior. This process is essential for ensuring the correct orientation

of the anterior-posterior axis, and patterning of the adjacent embryonic

tissue. However, the mechanisms driving this migration are not clearly

understood. Indeed it is unknown whether the position of the future

anterior is pre-determined, or defined by the movement of the migrating

cells. Recent experiments on the mouse embryo, carried out by Dr.

Shankar Srinivas (Department of Physiology, Anatomy and Genetics) have

revealed the presence of multicellular ‘rosettes’ during AVE migration.

We are developing a comprehensive vertex-based model of AVE migration.

In this formulation cells are treated as polygons, with forces applied

to their vertices. Starting with a simple 2D model, we are able to mimic

rosette formation by allowing close vertices to join together. We then

transfer to a more realistic geometry, and incorporate more features,

including cell growth, proliferation, and T1 transitions. The model is

currently being used to test various hypotheses in relation to AVE

migration, such as how the direction of migration is determined, what

causes migration to stop, and what role rosettes play in the process.

Abstract: Nonlinear models have been widely employed to characterize the

underlying structure in a time series. It has been shown that the

in-sample fit of nonlinear models is better than linear models, however,

the superiority of nonlinear models over linear models, from the

perspective of out-of-sample forecasting accuracy remains doubtful. We

compare forecast accuracy of nonlinear regime switching models against

classical linear models using different performance scores, such as root

mean square error (RMSE), mean absolute error (MAE), and the continuous

ranked probability score (CRPS). We propose and investigate the efficacy

of a class of simple nonparametric, nonlinear models that are based on

estimation of a few parameters, and can generate more accurate forecasts

when compared with the classical models. Also, given the importance of

gauging uncertainty in forecasts for proper risk assessment and well

informed decision making, we focus on generating and evaluating both point

and density forecasts.

Keywords: Nonlinear, Forecasting, Performance scores.

Hairsine-Rose (HR) model is the only multi sediment size soil erosion

model. The HR model is modifed by considering the effects of sediment bedload and

bed elevation. A two step composite Liska-Wendroff scheme (LwLf4) which

designed for solving the Shallow Water Equations is employed for solving the

modifed Hairsine-Rose model. The numerical approximations of LwLf4 are

compared with an independent MOL solution to test its validation. They

are also compared against a steady state analytical solution and experiment

data. Buffer strip is an effective way to reduce sediment transportation for

certain region. Modifed HR model is employed for solving a particular buffer

strip problem. The numerical approximations of buffer strip are compared

with some experiment data which shows good matches.

Abstract: Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is a ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last twenty years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs.

Dislocation channel-veins and Persist Slip Band (PSB) structures are characteristic configurations in material science. To find out the formation of these structures, the law of motion of a single dislocation should be first examined. Analogous to the local expansion in electromagnetism, the self induced stress is obtained. Then combining the empirical observations, we give a smooth mobility law of a single dislocation. The stability analysis is carried our asymptotically based on the methodology in superconducting vortices. Then numerical results are presented to validate linear stability analysis. Finally, based on the evidence given by the linear stability analysis, numerical experiments on the non-linear evolution are carried out.

Brownian Molecular Motors are crucial for cell motility, muscle contraction or any other mechanical task carried out by proteins. After a short introduction to protein motors, I will talk about a numerical appraoch I worked on during the last months, which should enable us to deduct properties for a broad range of protein motors. A special focus should lie on the calculation of the eigenvalue spectrum, which gives insight to motors' stability.

The circulatory system is the most important and amongst the most complicated mechanisms in the human body. Consisting of the heart, the arteries and the veins, it is amply aided by a variety of mechanisms aiming to facilitate adequate perfusion of the body tissues at the appropriate pressure. On this talk I am focusing on the development of a computational model which relates patient specific factors (age, gender, whether someone is an athlete/smokes etc) and their effects on different vascular regions which ultimately determine the arterial pressure and the cardiac output.