Gaussian Process Modeling of a Functional Output with Information from Boundary and Initial Conditions and Analytical Approximations

ABSTRACT

A partial differential equation (PDE) models a physical quantity as a function of space and time. These models are often solved numerically with the finite element (FE) method and the computer output consists of values of the solution on a fine grid over the spatial and temporal domain. When the simulations are time-consuming, Gaussian process (GP) models can be used to approximate the relationship between the functional output and the computer inputs, which consists of boundary and initial conditions. The Dirichlet boundary and initial conditions give the functional output values on parts of the space-time domain boundary. Although this information can help improve prediction of the output, it has not been used to construct GP models. In addition, analytical solutions of the PDE derived by simplifying the PDE can often be obtained, which can help further improve performance of the GP model. This article proposes a Karhunen–Loève (KL) expansion-based GP model that satisfies the Dirichlet boundary and initial conditions almost surely, and effectively uses information from analytical approximations to the PDE solution. Real examples demonstrate the improved prediction performance achieved by using these sources of prior information. Supplementary materials for this article are available online.

KEYWORDS:

• Constrained functional response modeling
• Finite element simulations
• Principal components analysis

Supplementary Materials

• Appendices.pdf: This file contains Appendices A–F. Appendix A gives a proof that the maximum likelihood estimator of the covariance matrix in Conti and O’Hagan’s method does not exist when N > n. Appendix B derives a condition for separability of the covariance function for the KL-GP model. Appendix C proves the interpolation property of the KL-GP model. Appendix D gives a proof of Theorem 1. Appendix E derives the asymptotic rates of convergence of $\mathit{S}$ and $\bar{\mathit{Y}}$. Appendix F gives an example that uses initial and Dirichlet boundary conditions to improve modeling of the output.

• Online ZIP file: This file contains Matlab codes for reproducing all reported numerical results.

Acknowledgments

The author thanks the editor, associate editor, two referees, and Marcus Hwai Yik Tan at Apple, California for valuable comments that helped improve the article.

Funding

This research was supported by General Research Fund project 11226716 funded by the Research Grants Council of Hong Kong.