Applied Numerical Computing Dissertation Topics
Typically students start working on the dissertation in April and complete it by September. The dissertation is normally 60-80 pages in length and features a mix of mathematics, application area, and computing.
The following gives example topics with a list of titles of recent dissertations.
Physical Applied Mathematics
The aim is to examine practical and theoretical aspects of a physical problem--- such as granular flow, wound healing, flame propagation---which are not well understood. The approach will typically adopt a combination of mathematical and numerical techniques (solution of ODEs or PDEs). The multi-disciplinary experience will be valuable for tackling research problems in other fields of application.
Recent dissertations in this area
- Numerical sudy of the interaction between spherical premixed and non-premixed flames
- Mathematical modelling of wound healing
- Measuring mass flow by conductive and convective transport of thermal energy
- Stochastic models of excitable media
Inverse Problems
Inverse Problems typically involve the recovery of some physical characteristics of a material from its response to some excitation such as electromagnetic fields, heat or mechanical vibrations. Inverse problems are often ill posed, which means the process of recovering the unknown parameters is very sensitive to errors in the measured response.
A potential project in this area may involve studying a new method for capacitance tomography using numerical solutions of PDEs and a little functional analysis. We have good relations with a number of companies working on this including Shell and Schlumberger (both employ mathematicians).
Recent dissertations in this area
- Measuring ocean current from a towed seismic array at real time
- Electrical impedance tomography and adaptive meshing
- The anisotropic inverse conductivity problem
- Linear Sampling
Mathematical Finance
Many interesting mathematical questions arising in pricing financial derivatives and options by way of stochastic differential equations and PDEs. It is rare to have exact solutions to price the options and computational techniques are required to compute prices for large portfolios efficiently. An example project is to study the pricing of barrier options, which involves computing times of exit of solutions of stochastic differential equations from a domain.
Recent dissertations in this area
- Application of Bayesian learning procedures to option pricing under stochastic volatility
- Trinomial tree discretisation of nonlinear Black Scholes equations
Image Processing
Computers are now widely used for the storage and analysis of images (such as images of the brain) and this involves sophisticated mathematical techniques in linear algebra approximation theory, optimisation, and differential equations to name a few. One standard problem in image processing is the recovery of a true image from an observed image, which may be blurred or noisy. Numerical methods based on PDEs have recently been developed for the image reconstruction. One project is to study efficient methods for solving the PDEs, which often lead to badly conditioned linear system when discretised.
Recent dissertations in this area
- Linear Algebra for Image Deblurring/Denoising Problems
- Optimizing the paths for Geodesic Interpolating Splines
- Data compression with the SVD
- Inpainting
Finite Element Methods and High Performance Computing
Highly stretched grids are needed if fluid flow problems are to be solved modelling flow around objects, for example, aeroplanes and racing cars. One possible project, theoretical in nature, will consider the stability, or otherwise, of some standard mixed finite element approximation methods. The project will introduce the student to the important field of mixed finite element approximation methods. Such methods arise in many applications, including modelling flow of an incompressible fluid.
Dissertations in this area are regularly offered by the Supercomputing, Visualization and e-Science group within Manchester Computing.
Recent dissertations in this area
- Parallelisation of least square algorithms
- A finite element approximation of non-Newtonian flows
- Improving and removing communications in a suite of parallel finite element analysis codes
- Object-Oriented Mesh Generation
Numerical Solution of Differential Equations
One project in this area is to develop numerical methods and computer software for solving initial value problems consisting of differential algebraic equations with delay terms. Questions to be investigated include: How do the delay terms affect the order of convergence and stability of the numerical methods? When are advanced delays likely to arise and how can they be handled? How do derivative discontinuities propagate between the differential equations and algebraic constraints?
Recent dissertations in this area
- Variable step collocation method to solve second order initial value problems
- Nonlinear equation solvers and the numerical solution of ordinary differential equations
Numerical Linear Algebra
Many opportunities are available to study pure and applied aspects of numerical linear algebra, which is at the heart of all numerical methods. One example is the quadratic eigenvalue problem
which arises in structural mechanics, with M the mass matrix, D the damping matrix and K the stiffness matrix. Numerical methods for this problem are currently a very active area of research.
A related problem is the nonsymmetric tridiagonal eigenvalue problem
where T is nonsymmetric tridiagonal, which arises as intermediate steps in a variety of eigenvalue problems. For example, the nonsymmetric eigenvalue problem can be reduced in a finite number of steps to nonsymmetric tridiagonal form. In the sparse case, the nonsymmetric Lanczos algorithm produces a nonsymmetric tridiagonal matrix.
Recent dissertations in this area
- MATLAB toolbox for classical matrix groups
- Computing nearest covariance and correlation matrices
Dynamical Systems
There are a number of important numerical measures used to classify the behaviour of dynamical systems, derived either from the solution of systems of differential equations or reconstructed from time series of experimental observations. While there has been a considerable body of work in the investigation of low-order systems (typically in three or less dimensions in phase space), many systems of interest evolve in a higher dimensional phase space. Recent dissertations related to this topic have developed an object-oriented code base for estimating the dimensions of an attractor in higher dimensional phase space. There are a number of highly interesting questions yet to be pursued, related to questions of parallelisation, efficient numerical techniques and the application of object-oriented programming languages to numerical computing.
The applications of the methods are related to problems in astrophysical fluid dynamics, in particular the dynamo generated magnetic fields observed in the sun and other stars and investigation of their complex behaviour in time and space. The code base is in C++ allowing students to gain skills in one of the most widely used programming languages for scientific and financial programming. Many of the ideas investigated in the analysis of nonlinear dynamical systems have wide application in sciences of complex systems, such as social, economic and biological systems. Recent dissertation topics have included the investigation of the geometry of chaotic systems via object-oriented numerical methods and the stability of the numerical measures (e.g. fractal dimension) to transformations in the symmetry group of the Euclidean plane (rotation, reflection and translation).
Recent dissertations in this area
- Parallelisation of methods for estimating the dimension of attractors
- Analysis of high-dimensional dynamical systems using phase-space embedding
- Geometrical methods for investigating chaotic attractors
Numerical Optimisation
Optimisation problems arise in all areas of science and engineering and their importance has led to much development of theory and algorithms. Many open issues remain , such as finding global minima, instead of the easier problem of local minima. This is especially important in problems where the objective function features small scale oscillations. Other interesting areas are optimisation in infinite dimensions, such as the optimisation problems that in image processing.
Recent dissertations in this area
- Symplectic integrators and optimal control
- Finding escape trajectories by the minimum action method
- Algorithms for global optimisation