PARTIAL Analysis of Relations in Tasks of Inversion for Algorithmic Leverage

The solution of inverse problems forms a crucial part of many modern technologies, such as medical and astronomical imaging devices. We want to obtain from possibly very limited and corrupt measurements, a good description of the phenomenon under study. Often this description takes the form of an image, as we humans like to process visual information, but it can also take many other forms, such as the location of an earthquake, the source of a sound, or the detection of a planet.

As physical technologies improve, and become more widespread, we are faced with increased amounts of data, as well as the need to process them into higher-quality results, such as higher-resolution images. We would also like to produce the results with as little energy as possible, and ideally in real time, instead of waiting computers to crunch numbers for hours after the measurement was taken. For this we need really good optimisation algorithms: instructions for a computer to find the point where a mathematical function reaches its smallest value, akin to the lowest part of a valley among mountains. This is because the solutions we are looking are such minimising points of suitable functions modelling our inverse problem.

The objective of the proposed project is to develop efficient optimisation algorithms for the solution of large-scale inverse problems. This will be based on a study to gain a better understanding into the stability of the solutions to these problems under perturbations of the measurements and the parameters of the model. Besides inverse problems, our work will be applicable to a diverse range of disciplines from financial planning to machine learning.

The solution of large-scale optimisation problems forms a crucial part of many modern technologies, including financial planning, autonomous and intelligent devices based on machine learning, as well as modern imaging devices from medicine and industry to astronomy and security. In imaging, optimisation surfaces through the need to solve an inverse problem to produce the image from limited or corrupt measurements in a non-image measurement domain.

As the involved physical technologies improve-e.g., devices for magnetic resonance imaging, electrical impedance tomography, computed tomography, electron and optical microscopes-and their use becomes more widespread, we are faced with increased amounts of data, as well as the need to process the data into higher-quality results, such as higher-resolution images, or an otherwise better understanding of the world. We would also like to produce the results with as little energy as possible, and fast, ideally in real time, instead of waiting computers to crunch numbers for hours after the measurement was taken. For this we need really good optimisation algorithms.

The objective of the proposed project is to develop such efficient optimisation algorithms, with focus on the solution of large-scale and PDE-constrained inverse problems. The eventual beneficiaries of the project will include everyone who uses technologies incorporating their solution. From the enhancement of digital pictures taken in poor lighting conditions, to the processing and automated understanding of images coming from devices for medical, industrial, and security imaging, as well as satellites and space exploration probes, these technologies are ever more widespread. The beneficiaries therefore range from consumers to doctors, from scientists to patients, and from travellers to the industry. Eventually, the outcomes of the proposed project will therefore foster the health and quality of life of the population, and improve industrial processes.

Secondly, this project will benefit the UK's initiative on Big Data. While our focus is on optimisation methods for inverse problems, our work will be beneficial to the solution of a wider range of large-scale optimisation problems, from financial planning to machine learning, the latter having applications from personal assistants to automated medicine. Our work will benefit a plethora of areas of modern society. Moreover, randomised algorithms form the very basis of quantum computing. As quantum computers become practical in the coming years, the stochastic aspects of the our research are likely to reach their full potential. In this way the project will help create new technologies, companies, and wealth in both the near and the more distant future.