A post by Conor Crilly, PhD student on the Compass programme.

Introduction

This project investigates uncertainty quantification methods for expensive computer experiments. It is supervised by Oliver Johnson of the University of Bristol, and is partially funded by AWE.

Outline

Physical systems and experiments are commonly represented, albeit approximately, using mathematical models implemented via computer code. This code, referred to as a simulator, often cannot be expressed in closed form, and is treated as a ‘black-box’. Such simulators arise in a range of application domains, for example engineering, climate science and medicine. Ultimately, we are interested in using simulators to aid some decision making process. However, for decisions made using the simulator to be credible, it is necessary to understand and quantify different sources of uncertainty induced by using the simulator. Running the simulator for a range of input combinations is what we call a computer experiment [1]. As the simulators of interest are expensive, the available data is usually scarce. Emulation is the process of using a statistical model (an emulator) to approximate our computer code and provide an estimate of the associated uncertainty.

Intuitively, an emulator must possess two fundamental properties

• It must be cheap, relative to the code
• It must provide an estimate of the uncertainty in its output

A common choice of emulator is the Gaussian process emulator, which is discussed extensively in [2] and described in the next section.

Types of Uncertainty

There are many types of uncertainty associated with the use of simulators including input, model and observational uncertainty. One type of uncertainty induced by using an expensive simulator is code uncertainty, described by Kennedy and O’Hagan in their seminal paper on calibration [3]. To paraphrase Kennedy and O’Hagan: In principle the simulator encodes a relationship between a set of inputs and a set of outputs, which we could evaluate for any given combination of inputs. However, in practice, it is not feasible to run the simulator for every combination, so acknowledging the uncertainty in the code output is required. (more…)