# Enabling Quantification of Uncertainty for Large-Scale Inverse Problems (EQUIP)

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

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.

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.

### Planned Impact

The work concerns the solution of inverse problems, and in particular the use of methodologies which allow for the quantification of uncertainty in such problems. It will be guided by applications arising in determining subsurface rock properties, including oil and gas extraction, nuclear waste burial, groundwater flow, carbon capture and storage (CCS) and compressed air storage for intermittent energy sources. Together these applications form significant components in the overall UK energy portfolio at a time when energy provision is undergoing significant change. Furthermore, the general problem of quantifying uncertainty in inverse problems has wide-ranging applicability in many problems arising in the biological, physical and social sciences; examples include neuroscience, climate prediction and adaptive control of traffic flow.

An Advisory and Impact board has been structured to help maximize the potential for impact. This board will contain senior academic, governmental and industrial stakeholders, will play an active role in guiding the evolution of the research over its entire duration and will help identify pathways to impact for the research undertaken, and for the research which will result, in the medium-term, from the creation of new methodologies for uncertainty quantification in the solution of inverse problems.

The research team will maintain its strong record in premier journals and at conferences, will create public or open-source versions of their codes and will examine the feasibility of full commercialisation. The team will also run bi-yearly industry-facing events, modelled on the succesful Industry Days at HW, but tailored to the specific research agenda arising from uncertainty quantification in inverse problems. The team will also organize two workshops to which leading EPSRC-funded UK researchers in areas such as CCS, compressed air storage, ground and water engineering and nuclear waste disposal will be invited to speak, and at which the new work will be presented. The research team will also seek direct engagement with the public through publishing lay scientific articles via a variety of UK and international scientific societies and will use press coverage to inform the public and industry of their work. Finally the research activity as a whole will act as a draw for other researchers, attracting them to the field; in particular research students from Heriot-Watt, UCL and Warwick and in particular from EPSRC funded ventures such as CRISM and the Complexity and MASDOC doctoral training centres at Warwick will be actively recruited to work in developing new methodologies, or their application.

An Advisory and Impact board has been structured to help maximize the potential for impact. This board will contain senior academic, governmental and industrial stakeholders, will play an active role in guiding the evolution of the research over its entire duration and will help identify pathways to impact for the research undertaken, and for the research which will result, in the medium-term, from the creation of new methodologies for uncertainty quantification in the solution of inverse problems.

The research team will maintain its strong record in premier journals and at conferences, will create public or open-source versions of their codes and will examine the feasibility of full commercialisation. The team will also run bi-yearly industry-facing events, modelled on the succesful Industry Days at HW, but tailored to the specific research agenda arising from uncertainty quantification in inverse problems. The team will also organize two workshops to which leading EPSRC-funded UK researchers in areas such as CCS, compressed air storage, ground and water engineering and nuclear waste disposal will be invited to speak, and at which the new work will be presented. The research team will also seek direct engagement with the public through publishing lay scientific articles via a variety of UK and international scientific societies and will use press coverage to inform the public and industry of their work. Finally the research activity as a whole will act as a draw for other researchers, attracting them to the field; in particular research students from Heriot-Watt, UCL and Warwick and in particular from EPSRC funded ventures such as CRISM and the Complexity and MASDOC doctoral training centres at Warwick will be actively recruited to work in developing new methodologies, or their application.

### Publications

Agapiou S
(2014)

*Analysis of the Gibbs Sampler for Hierarchical Inverse Problems*in SIAM/ASA Journal on Uncertainty Quantification
Agapiou S
(2017)

*Importance Sampling: Intrinsic Dimension and Computational Cost*in Statistical Science
Agapiou S
(2014)

*Bayesian posterior contraction rates for linear severely ill-posed inverse problems*in Journal of Inverse and Ill-posed Problems
Agapiou S
(2018)

*Unbiased Monte Carlo: Posterior estimation for intractable/infinite-dimensional models*in Bernoulli
Agapiou S
(2018)

*New Trends in Parameter Identification for Mathematical Models*
Bazargan H
(2015)

*Surrogate accelerated sampling of reservoir models with complex structures using sparse polynomial chaos expansion*in Advances in Water Resources
Bertozzi A
(2018)

*Uncertainty Quantification in Graph-Based Classification of High Dimensional Data*in SIAM/ASA Journal on Uncertainty Quantification
Beskos A
(2015)

*Sequential Monte Carlo methods for Bayesian elliptic inverse problems*in Statistics and Computing
Beskos A
(2017)

*Geometric MCMC for infinite-dimensional inverse problems*in Journal of Computational Physics
Briol
(2015)

*Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees*in arXiv e-prints