Automating optimisation subject to partial differential equations on high-performance computers.

Lead Research Organisation: University of Oxford
Department Name: Mathematical Institute

Abstract

Optimisation problems appear across all areas of engineering. Optimisation consists of maximising the
performance or minimising the cost of a system, subject to some constraints. For example, an aeronautical engineer will want to choose the best shape for a wing to maximise its efficiency, subject to the constraint that the wing will lift the aircraft, while a civil engineer will want to design the cheapest bridge that will support its load.

An important class of optimisation problems is where the constraint is given by the laws of physics, such as the physical laws for fluids (in the wing case) and structures (in the bridge case). These problems can be very hard, and usually require massive supercomputers to solve them. A significant amount of mathematical research has gone into investigating techniques for solving them.

Engineers currently face a major practical difficulty when trying to solve new kinds of such optimisation problems. The software required to solve these is very intricate, and often takes months or years to develop. This poses a very formidable barrier. This matters a lot, because these problems appear everywhere in engineering, and if we could solve them then we could design many things in a better way.

I propose to do this by developing a software framework to generate optimisation codes, rather than have engineers develop them by hand. While the optimisation software is very complex, it has a compact mathematical structure: I propose to generate the optimisation software from a simple high-level description of this mathematical structure. By generating the necessary software, engineers can spend their time on using it to solve real problems. This framework will provide engineers with the necessary optimisaton software in days or weeks instead of months or years.

Generating the optimisation codes from simple high-level input has another major advantage. The high-performance supercomputers necessary to solve these optimisation problems are extremely difficult to program efficiently, and are changing rapidly. Code must be tailored for a particular hardware architecture. As each new kind of computing platform comes out, an engineer must adapt the code. Instead, with my new approach, the engineer can simply re-generate the code from the same mathematical input, and the framework will specialise the code to best exploit the different platform. By updating the framework once, many engineers working on many different codes in many different areas can benefit quickly from advances in computational hardware.

I will apply the software developed to two important engineering problems.

The first engineering problem is found in the design of marine turbine farms for renewable energy. Marine renewable energy is very important to the UK. The government predicts that the industry will be worth £76 billion to the UK economy by 2050. A major problem facing the industry is how to position the turbines to extract the maximum possible energy from the tide. Choosing the best design is very important, as it can greatly change the efficiency. Solving this problem will directly contribute to the UK's energy security and carbon reduction goals.

The second engineering problem is identifying regions of the heart that are damaged (ischaemic). Ischaemic heart disease is the most common cause of death in Western countries. When a doctor suspects that a patient has ischaemia, it would be very beneficial to know its location and extent. One possible approach to rapidly identify ischaemia is to extract information from electrocardiograms (ECGs). The optimisation problem is to identify the ischaemia that best explains the ECG measured from the patient. Solving this problem will directly contribute to better healthcare decisions, reducing the mortality rate and improving the long-term prognosis of survivors.

Planned Impact

Emerging industry: this fellowship will have a major and direct impact on a crucial emerging
industry in the UK, the marine renewables industry (cf. MeyGen letter of support). The problem I
propose to tackle is described by them as their current "main project challenge". If funded, this
will have a direct impact on the efficiency of their proposed turbine farm designs, the economic
viability of their projects, and the encouragement of the future growth of the industry in the UK and
abroad. By equipping UK industry with a powerful tool for farm design, this project will give UK
companies a strong competitive edge over similar businesses in other countries.

Established industry: the ultimate aspiration for the cardiac application is its commercial
development as a clinical diagnostic machine. This would have an impact on the healthcare technology
sector. The marine renewable application will have a general impact on the health of UK industry by
assisting in the provision of clean, reliable, renewable energy. The framework itself will have a
wide range of impacts across the entirety of industrial engineering: optimisation problems are
ubiquitous across engineering, and any technology which assists in their solution will lead to more
efficient and less polluting engineering designs.

Academic beneficiaries: there will be significant benefit to the academic engineering, mathematical
optimisation, high-performance computing, marine engineering, cardiological and geoscientific
academic communities. The framework developed in this proposal will enable them to solve problems
currently too costly to attempt. This is more fully described in the section on academic
beneficiaries.

Public sector: the Government will benefit through the impact of this proposal on its energy
security and carbon reduction targets. Government regulators facing the difficult decision of
balancing electricity demands against environmental impacts will be given a powerful tool to assist
in their deliberations: with the ability to predict the power generated by a particular farm design,
more informed decisions can be made. The NHS will benefit from the cardiac application through
improved healthcare outcomes.

General public: the general public will benefit significantly through the provision of clean
renewable energy, the reduction of carbon dioxide emissions to the environment, the improvement of
the healthcare of acute coronary syndrome, the gain in jobs in the marine renewable sector, and the
improvement of the competitiveness of UK industries.

Publications

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Beskos A (2017) Geometric MCMC for infinite-dimensional inverse problems in Journal of Computational Physics

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Chapman S (2017) Analysis of Carrier's Problem in SIAM Journal on Applied Mathematics

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Chapman SJ (2017) Analysis of Carrier's problem in SIAM Journal on Applied Mathematics

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Charalampidis E (2018) Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with deflated continuation in Communications in Nonlinear Science and Numerical Simulation

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Croci M (2018) Efficient White Noise Sampling and Coupling for Multilevel Monte Carlo with Nonnested Meshes in SIAM/ASA Journal on Uncertainty Quantification

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E. Rognes M (2017) cbcbeat: an adjoint-enabled framework for computational cardiac electrophysiology in The Journal of Open Source Software

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Farrell P (2014) A Framework for the Automation of Generalized Stability Theory in SIAM Journal on Scientific Computing

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Farrell P (2019) Deflation for semismooth equations in Optimization Methods and Software

 
Description The main key finding of this research is that by taking the correct mathematical abstractions, software for PDE-constrained optimisation problems can be developed orders of magnitude more rapidly than was possible before.

Another key finding is that it is crucial to embed knowledge of the functional analytic structure of a problem into the algorithms for its solution; this is an exciting area of mathematical research that continues to this day.

A final key finding, one not anticipated in the original proposal, is that it is possible to devise deflation algorithms that robustly compute _multiple local minimisers_ of PDE-constrained optimisation problems.
Exploitation Route I have a PhD student (funded on the EPSRC CDT in PDEs at Oxford) whose central topic is the study of deflation algorithms for computing multiple local minimisers of topology optimisation problems, an important and ubiquitous task in engineering.

With collaborators, we will apply the results of this research to engineering design problems in biomedicine, construction, energy and other fields.
Sectors Aerospace, Defence and Marine,Chemicals,Construction,Digital/Communication/Information Technologies (including Software),Electronics,Energy,Environment,Manufacturing, including Industrial Biotechology,Transport

 
Description The software developed in this proposal was used by MeyGen for the design of a tidal stream project of up to 398MW at an offshore site between Scotland's northernmost coast and the island of Stroma. The software developed in this proposal was used by London Computational Solutions for the shape optimisation of Formula 1 racing cars.
First Year Of Impact 2013
Sector Aerospace, Defence and Marine,Energy
Impact Types Cultural,Economic

 
Description EPSRC Platform Grant (PRISM renewal)
Amount £1,971,628 (GBP)
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 05/2018 
End 05/2023
 
Description Embedded Computational Science and Engineering Support
Amount £60,299 (GBP)
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 09/2014 
End 05/2015
 
Description Industrial funding for a PhD student
Amount £58,000 (GBP)
Organisation Simula Research Laboratory 
Sector Academic/University
Country Norway
Start 10/2016 
End 10/2019
 
Description Industrial support for a PhD student
Amount £58,000 (GBP)
Organisation London Computational Solutions 
Sector Private
Country United Kingdom
Start 10/2016 
End 10/2019
 
Description Industrial support for a PhD student
Amount £58,000 (GBP)
Organisation Petrotechnical Data Systems (PDS) 
Sector Private
Country Netherlands
Start 10/2016 
End 10/2019
 
Description Lord Leigh Fund
Amount £6,000 (GBP)
Organisation University of Oxford 
Sector Academic/University
Country United Kingdom
Start 11/2014 
End 09/2018
 
Description PhD studentship from the EPSRC Centre for Doctoral Training in Partial Differential Equations
Amount £58,000 (GBP)
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 10/2017 
End 10/2020
 
Description PhD studentship from the EPSRC Centre for Doctoral Training in Partial Differential Equations
Amount £58,000 (GBP)
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 10/2017 
End 10/2020
 
Title Defcon 
Description Defcon is a tool for exploring the solutions of nonlinear partial differential equations. It is capable of computing solutions to differential equations that are exceptionally difficult to find by other means. 
Type Of Technology Software 
Year Produced 2017 
Open Source License? Yes  
Impact I have discovered new solutions to several systems of physical interest (in quantum mechanics and in liquid crystals) that were previously unknown. 
URL https://bitbucket.org/pefarrell/defcon
 
Title OpenTidalFarm 
Description OpenTidalFarm is an open-source software for simulating and optimising tidal turbine farms. 
Type Of Technology Software 
Year Produced 2017 
Open Source License? Yes  
Impact It has been used in industry by MeyGen, Atlantis & others. 
URL https://opentidalfarm.readthedocs.io/en/latest/
 
Title dolfin-adjoint 
Description dolfin-adjoint enables the automated derivation and solution of adjoint models associated with physical models. With EPSRC's funding, I have been able to develop a significant automated PDE-constrained optimisation capability (optimising designs subject to physical laws). This allows companies and scientists to solve problems in weeks that previously would have taken months or years. 
Type Of Technology Software 
Year Produced 2014 
Open Source License? Yes  
Impact * I am using it to optimise the design of arrays of tidal turbines for MeyGen Ltd (renewable energy). * I am in initial discussions with VerdErg Ltd to use it to improve the design of their renewable energy device. This will be the subject of a future proposal to Innovation UK. 
URL http://dolfin-adjoint.org
 
Description 2014 ANADE Summer School 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact I received excellent feedback from the summer school organisers: of all of the sessions of the summer school, they enjoyed mine the most.

I made contacts with some potential colleagues at the University of Cambridge who are interested in applying my software to shape optimisation of aeronautical designs.
Year(s) Of Engagement Activity 2014
 
Description 2014 Simula Summer School 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact There was a lot of discussion on possible applications of adjoints in biomedical applications.

There was extensive interest in the software that I am developing with EPSRC's support, especially for applications in cardiac electrophysiology.
Year(s) Of Engagement Activity 2014
URL https://www.simula.no/education/ssri/summerschool
 
Description 2016 Summer School with an SME 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact I was contracted by an SME (the Computational Modelling Initiative) to give summer schools on computational mathematics in Chicago and Paris.
Year(s) Of Engagement Activity 2015,2016,2017
 
Description Invited lecture at the AGM of an SME 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Industry/Business
Results and Impact I was asked to give the invited lecture at the Annual General Meeting of NAG Ltd, the main international consultancy firm in mathematical modelling and numerical analysis. My scientific lecture sparked an interesting debate about how equation solvers should be designed. NAG are interested in implementing some of the algorithms I have developed in their flagship product, the NAG library.
Year(s) Of Engagement Activity 2015