Contributions to Multi-Fidelity Gaussian Process Modeling and Emulator-Based Calibration of Computer Models

Abstract

This thesis develops a novel Gaussian process (GP) emulator for multi-fidelity simulations and two novel model calibration methods based on improved GP emulators built with data from time-consuming multi-fidelity simulations. It consists of five chapters. Chapter 1 introduces the problems solved in the thesis. The remaining chapters are summarized below.

Multi-fidelity simulations are widely employed in engineering. When the simulators are time consuming to run, an autoregressive GP (AGP) model fitted with data from a nested space-filling design can be employed as emulator. However, the AGP model assumes the simulators at different levels of fidelity share the same inputs. Chapter 2 considers bi-fidelity simulations with a high-fidelity (HF) simulator and a low-fidelity (LF) simulator, where the HF simulator contains a vector of inputs not shared with the LF simulator, called augmented input. The augmented input captures finer modeling details neglected by the LF simulator, and the HF simulator reduces to the LF simulator when some or all components of the augmented input tend to zero. To ensure this boundary constraint in the domain of the augmented input is satisfied, we propose a modified AGP model that uses covariance functions (CFs) constructed from covariances of integrated stochastic processes, called integrated CFs. Five families of integrated CFs are compared in two numerical examples based on finite element simulators and in numerical simulations based on four test functions with analytical forms. It is demonstrated that certain choices of integrated CFs yield substantial improvements in prediction performance attained by the modified AGP model over the standard AGP model.

HF simulators with parameters that need to be calibrated are common in engineering. To overcome computational challenges in calibrating HF simulators that yield high dimensional outputs and incur long run times, which are prevalent in practice, Chapter 3 proposes an efficient method to calibrate such simulators that uses a novel bi-fidelity GP emulator for predicting the sum of squared errors (SSE) between the HF simulator output and field data, called the HF SSE. The proposed bi-fidelity emulator fuses information in data from both the HF simulator and a faster LF simulator to reduce its need for costly HF simulator runs. It achieves this by linking a Box-Cox transformation of the HF SSE and the same transformation of a modified version of the SSE between the LF simulator output and field data, called the LF SSE. The LF SSE is modified so that it approximates the HF SSE better while the Box-Cox transformation is applied to the HF SSE and modified LF SSE to obtain transformed SSEs that are well modeled by a priori stationary GPs. The proposed emulator is used to estimate calibration parameters by minimizing the HF SSE via a Bayesian optimization method. A simulated example and an example on calibrating an office thermal environment simulator show that the proposed method outperforms several alternative methods.

In Chapter 4, we consider calibrating a HF simulator using data from bi-fidelity simulations in which boundary information of the form stated in Chapter 2 is available. In this case, since the HF output converges to the LF output when some or all components of the augmented input tend to zero, it follows that the HF SSE converges to the LF SSE when some or all components of the augmented input tend to zero, which is boundary information expressed in terms of the HF SSE and LF SSE. The AGP model for a Box-Cox transformation of the HF SSE and the same transformation of the modified LF SSE proposed in Chapter 3 and other existing bi-fidelity GP emulators for performing model calibration do not incorporate boundary information. To exploit boundary information for model calibration, we model a Box-Cox transformation of the HF SSE and the same transformation of the LF SSE with the BMAGP model proposed in Chapter 2, which guarantees that boundary information is satisfied. We employ the proposed model to estimate calibration parameters by minimizing the HF SSE via a Bayesian optimization method. One simulated example and an example based on a pair of HF and LF simulators of a bimetallic cantilever beam are given to illustrate the improvements in model calibration performance attained by the proposed model over alternative models for predicting the HF SSE.

Chapter 5 concludes this thesis and discusses possible future work.

Date of Award	19 May 2023
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Matthias Hwai-yong TAN (Supervisor)