Topics in design of computer experiments

Abstract

Computer models are used in many fields to simulate real-world processes. One of the goals is to optimise the value of the computer model. Due to the fact that the computer model is usually expensive to evaluate, one can only make a limited number of evaluations of this computer model. Using Gaussian processes in sequential design through the use of acquisition functions is a common approach for sample-efficient optimisation in such cases. Despite many recent successes, there are still a number of outstanding problems in that field. In this work, we address some of these problems.

We first give a brief overview of Bayesian optimisation and the common techniques used. After that, we focus on scenarios where one is interested in finding both the minimum and the maximum of the computer model simultaneously. We use the entropy of the location of the optima to define a sequential design algorithm. The design is then created in a way that would minimise the entropy. Monte Carlo methods are used to approximate a number of probability distributions. The resulting algorithm is then compared against a baseline algorithm to demonstrate its superior performance.

In the second part of this paper, we are interested in optimising high-dimensional computer models. This is a complicated task and comes with a number of additional challenges compared to the standard problems. Our focus is on computer models whose accuracy we can control by changing the amount of computational resources allocated to them. This is also often referred to as multi-fidelity optimisation. We then use a number of techniques from mathematical optimisation to define a multi-fidelity optimisation algorithm that can be used in high-dimensional settings and scaled to large number of evaluations. Its performance is then compared to that of another state-of-the-art optimisation algorithm.

In the final part of this paper, we explore computer models that are non-stationary. These are computer models whose properties change depending on which part of the design space it is evaluated. For example, a computer model that changes rapidly in one area and is completely flat in a different area is non-stationary. Such computer models can accurately be modelled with a non-stationary Gaussian process. However, fitting non-stationary Gaussian processes can be computationally very expensive and a large number of computer model evaluations is often needed to model the process accurately. This makes sequential design difficult. We create a novel acquisition function that allows us to create accurate sequential designs by using regular stationary Gaussian processes that are far easier to fit. Our acquisition function puts more emphasis on more interesting regions and less emphasis on less interesting regions. We then create sequential designs created by our novel acquisition function combined with stationary Gaussian processes and compare it with designs found by non-stationary Gaussian processes.

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Identifiers

ORCID for Hendriico Merila: orcid.org/0009-0006-4693-1267

ORCID for Antony Overstall: orcid.org/0000-0003-0638-8635

ORCID for Dave Woods: orcid.org/0000-0001-7648-429X

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