Details of Grant 

EPSRC Reference:	EP/Y001028/1
Title:	Unifying Probabilistic Computation for PDEs and Linear Systems
Principal Investigator:	Cockayne, Dr JP
Other Investigators:
Researcher Co-Investigators:
Project Partners:	Eberhard Karls University of Tubingen Instituut voor Sterrenkunde
Department:	Sch of Mathematical Sciences
Organisation:	University of Southampton
Scheme:	Standard Research - NR1
Starts:	01 December 2023	Ends:	30 November 2025	Value (£):	142,852
EPSRC Research Topic Classifications:	Statistics & Appl. Probability
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 17 May 2023 ECR International Collaboration Grants Panel 1 Announced
Summary on Grant Application Form
This research programme focuses on accelerating computer models, particularly when they are incorporated into inverse problems. Computer models function by solving equations that model reality, such as linear systems or partial differential equations (PDEs). These can be used to model diverse processes, from electrical conductivity in the human heart to stellar evolution. However computer models are useless without estimates of their parameters, and these estimates are often obtained from data in a problem referred to as the inverse problem. Solving an inverse problem usually requires simulating from a model repeatedly for different values of the parameters. As a result it is highly computationally expensive, and is often a limiting factor in the complexity of models that can be used. To address this we will develop novel probabilistic numerical methods (PNMs) in close partnership with the University of Tuebingen. PNMs are numerical methods that return a probability distribution describing uncertainty due to discretisation error. A user can use a coarser discretisation to approximate the solution to the PDE or linear system faster, and obtain "wider" uncertainty quantification as a result, reflecting that the solver is less confident in the solution. Importantly, this uncertainty can be propagated rigorously into the solution of an inverse problem, so that parameter estimates reflect the level of accuracy in the solver. This allows the user to reduce the computational expense of solving the inverse problem while retaining statistically rigorous parameter estimates. In more detail, we will focus on solving two problems at the heart of probabilistic linear solvers and PDE solvers. In the former case, we will develop solvers that are faster and more accurate than existing routines, by acknowledging and correcting for a contradiction in the fundamental methodology of those solvers (namely, that the procedure employed does not acknowledge the fact that the data depend on the solution). For PDE solvers, we will focus on solving nonlinear PDEs, which are more challenging to solve and provide a more realistic description of physical phenomena, but have thus far eluded a rigorous probabilistic treatment. To demonstrate the impact of these solvers we will apply them to a challenging inverse problem in astrophysics with partners at KU Leuven: 3D deprojection of stars based on measurements of their stellar wind. The impact of this could be far reaching. Computer models are used widely in the applied sciences and industry, for example in manufacturing and engineering, biology and healthcare, and accelerating them could be a stepping stone to enabling more widespread use. At the same time, in the context of rising energy costs and chip shortages, reducing the cost of models that have already been deployed could provide major economic benefits.
Key Findings
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Organisation Website:	http://www.soton.ac.uk