## Introduction

Many data-centric engineering problems are ‘inverse’ in nature, i.e. they involve working out unknown parameters and causes from observations of a system of interest, rather than modelling a system from known parameters. For example constructing an image of someone’s internal organs from an x-ray. This project involves the improvement of the statistical modelling of these unknown parameters in a range of important contexts.

## Explaining the science

An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or modelling groundwater flow from geophysical measurements of the earth’s subsurface.

They are called inverse problems because they start with the results and then calculate the causes. This is the inverse of a forward problem, which starts with the causes and then calculates the results.

When faced with an inverse problem, the important question is how to make it computationally viable. Often the quantities you want to make inferences about are infinitely dimensional, in the form of a ‘function surface’ rather than just numbers. It’s therefore necessary to use statistical modelling to reduce the dimensions of parameter spaces and reduce the number of times you need to run computations.

The field requires mathematicians to describe the physics of a system, statisticians to deal with infinite dimensional nature of the unknown parameters, and computational people to run the calculations.

## Project aims

This project is aiming to provide efficient and accurate statistical solutions of inverse problems that are physics-constrained. This is being done by combining computational methods and related ‘probabilistic numerical methods‘ that aim to more accurately model the difference between a real, physical system and a mathematical approximation of that system.

Together, these tools enable the accurate quantification of uncertainty in inverse problems, in contexts such as providing statistical confidence regions for tissues in medical scans, or safely separating fluid mixtures with industrial hydro-cyclone equipment.