- Opens: Thursday 27 February 2020
- Number of places: 1
- Duration: 3 years
OverviewMany problems are described by partial differential equations that are too difficult to solve analytically. Numerical solutions must be used, which typically require the solution of linear systems. This project will look at how to solve these linear systems efficiently.
Applicants should have a good 2:1 degree in mathematics or a related discipline.
Knowledge of numerical analysis, and particularly the numerical solution of PDEs and/or iterative methods for solving linear systems is desirable but not essential.
Previous experience of a suitable programming language, e.g. Matlab, Python, Julia, C, is desirable but not essential.
To solve real-world problems in, e.g., climate modelling, aircraft design, biological modelling, we often require the numerical solution of partial differential equations. The usual methods for this, which include finite difference, finite element, finite volume and radial basis methods, result in one or more linear systems that require solution.
In many cases these linear systems possess a special structure that makes products between the coefficient matrix and a vector fast. For such problems, iterative methods known as Krylov subspace methods, are typically used. These methods can be extremely fast, but only if the number of iterations is kept low. This project will investigate how to achieve this fast convergence for problems of interest.