Uncertainty Quantification for Numerical Models with two Regions of Solution

The use of complex numerical models in science and policy is now ubiquitous. For practical application predictions from such models need to be accompanied by estimates of uncertainty. Supplying these estimates is a difficult problem. Gaussian process emulation has proved to be a very effective solution A Gaussian process emulator is a statistical model of the numerical code that is fast to run so can be used to calculate uncertainty estimates, for example by Monte Carlo methods that are computationally impossible with the full model. Emulators have been used across a wide variety of applications from climate and air pollution through engineering and geology to systems biology. However there are numerical models where there are two or more solutions separated by bifurcations or tipping points. A simple example from climate science is the Stommel model which has a different solution for when the overturning circulation is turned on or off. We have other examples from climate, oil reservoir modelling and biology. This PhD will look at this problem developing methods to identify the areas of model input space where each solution holds and how to build separate, but linked, emulators for each solutions. The inverse problem where we have real world observations of some of the model outputs and these are used to make inferences about the model inputs, in effect running the model 'backwards', will also be considered. In addition the student will look at the problem of experimental design, where best to run the model with a limited computer budget, developing new sequential methods. Although the student will develop a general methodology it will be applied to a number of real world applications throughout the PhD.