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Department of Mathematics

Inverse problems

In inverse problems researchers look inside solid objects or deduce complex models from data using mathematics.

Inverse problems typically involve the recovery of some unknown parameters of a system from its response to an excitation such as electromagnetic fields, heat, or mechanical vibrations. Inverse problems are often challenging since they are ill-posed, which means the process of recovering the unknown parameters is very sensitive to errors in the measured response. Mathematical methods in inverse problems confront this ill-posedness through regularisation or statistical techniques.

We look at both practical and theoretical aspects of the subject, and our work has links with many other areas of mathematics, both pure and applied, as well as science, engineering and medical research. Many of our projects are interdisciplinary and produce open source computer code used by researchers in other disciplines.

We are actively engaged in research projects related to a wide range of topics, including:

  • Electrical impedance tomography with applications in medical imaging and industrial process monitoring.
  • Tomographic reconstruction with applications to X-ray and neutron tomography.
  • Next generation metal detectors for security screening and land mine detection.
  • Geophysical imaging.
  • Variational methods including regularisation techniques and convex optimisation.
  • Bayesian inverse problems.
  • Markov chain Monte Carlo, sequential Monte Carlo samplers and multilevel as well as multi-index Monte Carlo methods with applications in biology and material science.
  • Machine learning enhanced inversion methods.
  • Microlocal analysis and geometric inverse problems.

More information about our research outputs and research-related activities can be found by browsing the web pages of the staff listed on this page. Potential PhD students may email academic staff directly to discuss possible projects.

Connections to data science

Inverse problems as a discipline deals both with fundamental questions concerning what type or how much data is required to recover certain parameters or form images with a desired accuracy, as well as practical algorithms to pass from data to reconstructions.