A mathematical model for a physical experiment is a set of equations which relate inputs to outputs. Inputs represent physical variables which can be adjusted before the experiment takes place; outputs represent quantities which can be measured as a result of the experiment.

The forward problem refers to using the mathematical model to predict the output of an experiment from a given input. The inverse problem refers to using the mathematical model to make inferences about input(s) to the mathematical model which would result in a given measured output.

An example concerns a mathematical model for oil reservoir simulation. An important input to the model is the permeability of the subsurface rock. A natural output would be measurements of oil and/or water flow out of production wells. Since the subsurface is not directly observable, the problem of inferring its properties from measurements at production wells is particularly important. Accurate inference enables decisions to be made about the economic viability of drilling a well, and about well-placement.

In many inverse problems the measured data is subject to noise, and the mathematical model may be imperfect. It is then very important to quantify the uncertainty inherent in any inferences made as part of the solution to the inverse problem.

The work brings together a team of mathematical scientists, with expertise in applied mathematics, computer science and statistics, together with engineering applications, to develop new methods for solving inverse problems, including the quantification of uncertainty. The work will be driven by applications in the determination of subsurface properties, but will have application to a range of problems in the biological, physical and social sciences.