Abstract
Efficient surrogate modelling of computer models (herein defined as simulators) becomes of increasing importance as more complex simulators and non-deterministic methods, such as Monte Carlo simulations, are utilised. This is especially true in large multidimensional design spaces. In order for these technologies to be feasible in an early design stage context, the surrogate model (oremulator) must create an accurate prediction of the simulator in the proposed design space. Gaussian Processes (GPs) are a powerful non-parametric Bayesian approach that can be used as emulators. The probabilistic framework means that predictive distributions are inferred, providing an understanding of the uncertainty introduced by replacing the simulator with an emulator, known as code uncertainty. An issue with GPs is that they have a computational complexity of O(N3) (where N is the number of data points), which can be reduced to O(NM2) by using various sparse approximations, calculated from a subset of inducing points (where M is the number of inducing points). This paper explores the use of sparse Gaussian process emulators as a computationally efficient method for creating surrogate models of structural dynamics simulators. Discussions on the performance of these methods are presented along with comments regarding key applications to the early design stage.
Highlights
Computer models are a vital tool in exploring engineering design options
An additional strength of this approach is the statistical framework of a Gaussian Processes (GPs) emulator, which provides an assessment of code uncertainty [3] — uncertainty introduced by approximating the simulator with an emulator
The code uncertainty can be included within an optimisation setting by using a Bayesian optimisation approach [4, 5]; or in a calibration setting with methods such as Bayesian history matching [6, 7]
Summary
Computer models (simulators) are a vital tool in exploring engineering design options. Quiñonero-Candela and Rasmussen present a unified framework for model approximations [10] These approaches seek to modify the joint prior p (f∗, f ) of the GP (Eqn 3) in order to replace the complexity of inverting Kf,f with a less expensive inversion. This is performed by incorporating inducing points {Z, u} (where Z are a set of inducing inputs and u are the corresponding latent function evaluations) into the joint prior p (f∗, f , u) and marginalising the inducing variables, u, out of the posterior ( Z will affect the final solution). The posteriors q (f ∗ | y, θ) and log marginal likelihoods p (y | X) for the DTC and FITC approximations can be unified into the analytical form outlined in Eqn 8 and Eqn 9. [12]
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