Coverage

The Computational Mathematics research program actively studies theoretical aspects of computational algorithms, both in the continuous and discrete settings, as well addressing implementation issues to ensure efficient and reliable solution techniques.

The specific areas addressed by the program include:

• high-dimensional approximation and sparse grid methods
• efficient solution of large scale problems
• numerical linear algebra
• numerical solution of partial differential equations
• parallel numerical methods
• inverse problems
• optimisation techniques
• parameter estimation
• uncertainty quantification
• regularisation methods
• thin plate spline smoothing
• tsunami and flood modelling
• plasma theory and modelling

Seminar

The group runs a regular seminar (usually 4pm on a Monday afternoon). If you would like to be notified of these seminars and associated notifications consider joining our mailing list.

Past seminars

Courses

Each year the program runs graduate levle courses. For instance, in 2020 we ran the following:

Topic: Stochastic methods in computational mathematics with applications to data science

Lecturer: Lindon Roberts

Semester: 1

Pre-requisites: Math1116 or Math2305 essential. The course will require both theoretical and computational work, so students should be comfortable with proving mathematical theorems and willing to program (all programming will be in Python). To this end, some background in analysis, scientific computing, probability or programming would be advantageous (e.g. Math2320, Math3511/3512/3501, Math3029, or Comp1100/1730).

Topic: Computational Algebraic Geometry

Keywords: Algebraic Varieties, Polynomial Ideals, Hilbert's Nullstellensatz, Gröbner basis, polynomial homotopy continuation

Lecturer: Martin Helmer and Markus Hegland

Semester: Semester 1 

Pre-requisites: Advanced Algebra 1 (i.e. knowing what a polynomial ring is)

We will follow some combination of the following books:

-- Ideals, Varieties, and Algorithms (Cox, Little, O'Shea)
-- Invitation to Nonlinear Algebra (Michalek and Sturmfels)
-- The Numerical Solution of Systems of Polynomials Arising in Engineering and Science (Sommese and Wampler)

Topic: Theory and techniques for IBVPs

Lecturer: Kenneth Duru

Semester: 2

Pre-requisites: MATH1013/4, MATH2305/6, MATH3511 (optionally relevant MATH3501, MATH3015, ASTR3002)

Model problems will consist of advection and diffusion equations. Application problems will involve flow and wave propagation problems.The course will cover the fundamentals of the theory and numerical methods for time-dependent PDEs in bounded domains. It will begin with the well-posedness and stability theory of IBVPs at the PDE level.

 It will cover classical numerical methods  such as FV, FE and FD methods. It will also introduce modern methods such as DG and spectral element methods. There will be special attention on the mathematical tools and techniques, like Fourier methods (including classical von Neumann analysis), energy methods, to prove numerical stability and convergence.