Computational mathematics

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. In 2020 we plan 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.

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    Project Supervisors
    Adaptive sparse grids
    Advanced computational techniques
    Clustering techniques
    Computational applications of Multiple Region relaXed MHD (MRxMHD)
    Computational Methods in Real Algebraic Geometry and Applications
    Discontinuous Galerkin method for the shallow water wave equation using physics based numerical fluxes
    Domain decomposition/Multiscale physics
    Edge Localised Modes – linear stability and dynamics
    Energetic Particle Physics of the International Thermonuclear Experimental Reactor (ITER)
    Evaluation of hydrological models
    Exploring Adaptive Mesh Refinement strategies for dynamic earthquake rupture modeling with ExaHyPE
    Fault tolerant algorithms
    High dimensional approximation
    Navier Stokes equation with free boundaries
    Parallel high-dimensional density estimation
    Parallel optimisation algorithms for large-scale machine learning
    Particle orbits in magnetic islands and chaotic magnetic field
    Reduced models in Plasma Cylinder
    Regularised black-box optimisation algorithms for least-squares problems
    Scalable Fault-tolerant PDE Solvers
    Sensitivity Analysis of environmental models
    Shaping value of information to real world conditions in water decision making
    Symbolic and Numeric Computation in Algebraic Geometry
    Synthetic diagnostics for global computer networks and fusion power experiments
    The thermodynamics (and entropy) of redistribution of energetic ions due to wave-particle interaction
    Thin plate splines
    Topological Data Analysis for detecting consistent patterns of spread for extremist content
    Tsunami and flood modelling
    Upwind summation-by-parts (SBP) finite difference methods for 3D seismic wave propagation in complex geometries
    Using methods from algebraic geometry to develop numerical approximations