Between January and June 2018, the Isaac Newton Institute for Mathematical Sciences in the UK ran a six-month programme on uncertainty quantification. The aim of the programme was to bring together the applied mathematics/numerical analysis and the statistical communities, who have different approaches to the problem on quantifying uncertainty in complex numerical models. Despite joint initiatives from groups such as SIAM and the ASA on journals and conferences the two communities remain separate and there was little understanding from one group on what the other does. The programme was organised by Peter Challenor (University of Exeter), Max Gunzberger (Florida State University), Catherine Powell (University of Manchester) and Henry Wynn (London School of Economics). Our core themes were: surrogate models; multilevel, multi-scale, and multi-fidelity methods; dimension reduction methods; inverse UQ methods; and careful and fair comparisons. INI programme participants attend the programme for up six months and have opportunities to work together in a collaborative way. Most of our participants attended for between 2-4 weeks. In addition to the participants we had a number of workshops. Four one week workshops on: key UQ methodologies and motivating applications (an introductory workshop to give introductions to UQ methodologies from both traditions);, surrogate models for UQ in complex systems; reducing dimensions and cost for UQ in complex systems; and UQ for inverse problems in complex systems; and two one day workshops aimed at industry and other stakeholders. Most of the talks from the workshops are available on line. I will outline what happened during the programme and give my personal views on the achievements of the programme and what is still left to do.

1. The Isaac Newton
Institute Uncertainty
Quantification Programme
A Personal Perspective
Peter Challenor
University of Exeter

2. What is the Isaac Newton Institute
for Mathematical Sciences?

3. • Based in Cambridge, UK
• Visitor research institute
• 6-month programmes

4. What was the UNQ
Programme?
• Jan - Jun 2018
• Bring together applied mathematicians/numerical analysts and statisticians
working on uncertainty quantification.
• Five core themes
- Surrogate models
- Multilevel, multi-scale, and multi-fidelity methods
- Dimension reduction methods
- Inverse UQ methods
- Careful and fair comparisons

5. How the INI works
Participants and Workshops
• The UNQ programme was six months long from Jan -Jun
2018
• Programme participants stayed in Cambridge for 2 weeks
to 6 months
• We also held workshops that non-participants could
attend

6. Organisers
• Peter Challenor (University of Exeter)
• Max Gunzberger (Florida State University)
• Catherine Powell (University of Manchester)
• Henry Wynn (London School of Economics)

7. Workshops
1. Key UQ methodologies and motivating applications
2. Surrogate models for UQ in complex systems
3. Reducing dimensions and cost for UQ in complex
systems
4. UQ for inverse problems in complex systems
As far as possible we balanced speakers between the
applied maths and statistics communities

8. Workshop 1
Key UQ methodologies and motivating
applications

9. Workshop 2
Surrogate models for UQ in complex
systems

10. Workshop 3
Reducing dimensions and cost for UQ in
complex systems

11. Workshop 4
UQ for inverse problems in complex
systems

12. Turing Gateway
• Two meetings organised by the Turing Gateway
• One at the start
• One at the end
• Aimed at stakeholders and industry
• Each attracted about 80 participants.

13. Everything is On-Line
• Almost all the talks at the INI are recorded
• Most of the 6 workshops are on line
• Lots of seminars in addition
• The nerdiest, geekiest box set ever!! Over one months of
seminars
• http://www.newton.ac.uk/event/unq

14. Some Results
• What happened to the careful and fair comparisons?
• In almost all cases we don’t solve the same problems
• There are subtle, and no so subtle, differences in the way
we set up the problems

15. Differences between the applied
maths and statistical approaches
• Much of the work in applied maths is about uncertainty
arising from discretisation in the numerical solution of
PDEs
• Statistics has never really considered this problem
• Similarly statisticians do not do intrusive solutions (some
really good linear algebra in the app maths approaches)

16. Both approaches look at the uncertainty
arising from uncertain inputs
• Statistical approach:
• Treat model as an unknown function
• Model the unknown function as a random function
• Model the random function as a Gaussian process
• Use the emulator in place of the model

17. • Applied maths approach
• Look at expectations of QoI
• For polynomial chaos/stochastic computation use
quadrature based on orthogonal polynomials (depend on
form of f(x))
• Where to truncate the expansion?
• Sparse grids
• Put error bounds on the estimate of E(x) and truncate to get
within these bounds
E(x) =
∫
∞
−∞
g(x)f(x)dx
E(x) =
∞
∑
n=1
αiϕi(x)

18. Language
• Uncertainty/error
• Experimental design
• Using Bayes theorem doesn’t necessarily make you a
‘Bayesian’
• Silly things - is ‘x’ always a spatial variable?

19. The Meaning of Error
• What do we mean by uncertainty?
• Numerical analysts often give firm bounds. The error is less that x
• Statisticians use distributions
• How do we reconcile these?
• What does it mean if a give a probability of being outside a fixed
interval?
• Is there some distribution within an interval (uniform, triangular,
beta?)

20. Multi-level and Multi-
Fidelity Models
• Models of exist at different resolutions (multi-level)
• Or different levels of complexity (multi-fidelity)
• Or different models of same complexity (multi-???)
• Interesting different approaches from the different
communities (MLMC, multi-level emulators)
• Possibility of hybrid methods
• Trade off between bias and variance

21. Matching GP’s to PDE’s
• Our Gaussian Process emulators use correlation functions
from geo-spatial statistics
• Are these general purpose kernels the best to use?
• Can we tailor them to the problem?
• +ve, montone, convex functions (J-P Gosling; Olivier
Roustant)
• Constraints
• Gaussian processes that are the solution of PDEs

22. Dimension Reduction
• Common problem
• Input dimension reduction - Active subspaces
• Output dimension reduction - Salter and
Williamson(2017)

23. Non-stationarity
• J-P Gosling - Voronoi tessellations
• Louise Kimpton - Latent Gaussian processes (poster)
• Tom Santner
• Andrew Stuart
• ‘Deep Gaussian’ processes (Gramacy, Teckentrup,
Dunlop)

24. Model Discrepancy
• Difference between the ‘model’/simulator and the ‘real’
world
• Difference between the numerical solution and the
underlying, infinite dimensional PDE
• Difference between that PDE and the real world
• Review paper being written

25. Experimental Design
• Sequential design
• Interaction between real world and model world
experiments

26. Models to Decisions
• A network on decision making under uncertainty using
complex numerical models
• 3 themes
• UQ
• From models to decisions
• Communicating uncertainty
• www.models2decisions.org

27. Models to Decisions
Conference 2018

28. Research Agenda
• One of our outputs is a research agenda
• First draft from conference
• Recommendations for research in all three themes

29. UQ Recommendations
• How can we develop a rigorous mathematical framework for
treating model error?
• How can we quantify and manage uncertainty well when we
have chains/ensembles/networks of models?
• How can we reconcile probabilistic statements about
uncertainty with deterministic bounds for numerical error?
Can we combine them in meaningful ways?
• How can we design surrogate models that (i) have
guaranteed error control, (ii) satisfy important physical
constraints?

30. • How can we better fuse data (which is becoming increasingly
available) and models in UQ studies, and provide rigorous
underpinning mathematics?
• How can we deal with high-dimensional, time-varying and
heteroscedastic uncertain processes?
• While UQ is well established in applications like engineering, how can
we advance UQ methodology in newer applications in areas such as
biology, healthcare and finance?
• Can we use causality inferred from data to validate the form of the
model? This may be particularly important in the biological and social
sciences where there are no physical laws to guide model building.
• The use of UQ methods with data-based Machine Learning/Artificial
Intelligence models (and the wider use of ML/AI in UQ)

31. • See me if you would like to see there full research agenda
including the other two themes recommendations

32. Conclusions
• No general unified theory of UQ
• All methods are valid in their own area
• Need to understand how different methods work
• Hard - lots of different types mathematics + domain
science
•