A comparison of polynomial chaos and Gaussian process emulation for uncertainty quantification in computer experiments

Abstract

Computer simulation of real world phenomena is now ubiquitous in science, because experimentation in the field can be expensive, time-consuming, or impossible in practice. Examples include climate science, where future climate is examined under global warming scenarios, and cosmology, where the evolution of galaxies is studied from the beginning of the universe to present day. Combining complex mathematical models and numerical procedures to solve them in a computer program, these simulators are computationally expensive, in that they can take months to complete a single run.

The practice of using a simulator to understand reality raises some interesting scientific questions, and there are many sources of uncertainty to consider. For example, the discrepancy between the simulator and the real world process. The field of uncertainty quantification is concerned with the characterisation and reduction of all uncertainties present in computational and real world problems. A key bottleneck in any uncertainty quantification analysis is the cost of evaluating the simulator. The solution is to replace the expensive simulator with a surrogate model, which is computationally faster to run, and can be used in subsequent analyses.

Polynomial chaos and Gaussian process emulation are surrogate models developed independently in the engineering and statistics communities respectively over the last 25 years. Despite tackling similar problems in the field, there has been little interaction and collaboration between the two communities. This thesis provides a critical comparison of the two methods for a range of criteria and examples, from simple test functions to simulators used in industry. Particular focus is on the approximation accuracy of the surrogates under changes in the size and type of the experimental design. It is concluded that one method does not unanimously outperform the other, but advantages can be gained in some cases, such that the preferred method depends on the modelling goals of the practitioner. This is the first direct comparison of polynomial chaos and Gaussian process emulation in the literature.

This thesis also proposes a novel methodology called probabilistic polynomial chaos, which is a hybrid of polynomial chaos and Gaussian process emulation. The approach draws inspiration from an emerging field in scientific computation known as probabilistic numerics, which treats classical numerical methods as statistical inference problems. In particular, a probabilistic integration technique called Bayesian quadrature, which employs Gaussian process emulators, is applied to a traditional form of polynomial chaos. The result is a probabilistic version of polynomial chaos, providing uncertainty information where the simulator has not yet been run.