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Department of Mathematics

Student using computer in computer cluster

Uncertainty quantification

Uncertainty quantification (UQ) is a modern inter-disciplinary science that combines statistics, numerical analysis and computational applied mathematics.

Today, UQ is a broad term used by diverse scientific communities to describe methodologies for taking account of uncertainties when mathematical and computer models are used to estimate quantities of interest and make predictions related to real-world processes.

When we simulate real-world phenomena (eg fluid flows, the spread of infections, the weather) using mathematical models, there is always uncertainty in our predictions. Some of this stems from the errors that we make when we use numerical methods on computers to approximate solutions to complex models. However, we also frequently encounter model uncertainty. A lack of knowledge about the underlying processes and their scales means that we can only ever adopt models that reflect our best understanding of reality. In practice, we are also restricted to using models that can be solved with available computing resources, limiting accuracy. Model inputs are also often uncertain, because they cannot be measured or else are only partially or indirectly observed.

The goal in UQ is to provide an estimate, often in terms of a probability, of a quantity of interest related to the inputs or outputs of a model. Hence, UQ includes not only the development and analysis of numerical algorithms for propagating uncertainty from model inputs to outputs, but also for solving inverse problems, the study and analysis of uncertainty in models themselves, stochastic modelling techniques, and much more.

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

  • Approximation theory and error estimation for PDE models with random inputs
  • Efficient adaptive and multilevel algorithms for forward and inverse UQ
  • Statistical and Bayesian inverse problems
  • Markov Chain Monte Carlo (MCMC) methods
  • Stochastic and multiscale modelling
  • Model emulation and calibration
  • Design of efficient experiments for UQ
  • Applications in engineering, geophysics, life sciences, epidemiology and public health

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

Connections to data science

Although the quantity and complexity of data available to researchers continues to increase in many application domains, there are many important scenarios in science and engineering where there is a lack of data, leading to uncertainty. Mathematical and statistical tools that make the best use of limited data to make predictions, and that can inform us how best to gather more data (if possible) in order to gain improved estimates of quantities of interest, are essential.

UQ intersects with data science in many ways. A natural example is in the numerical solution of Bayesian inverse problems, where there is a need to develop statistical sampling methods to efficiently estimate posterior distributions of uncertain model inputs. As more and more data becomes available, developing hybrid approaches to modelling that combine classical mechanistic models with new data-driven and machine learning techniques is also an important challenge.