Gaussian functional regression for linear partial differential equations
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functional_Gaussian_processes_revised.pdf
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2.19 MB
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Adobe PDF
Checksum (MD5)
4a8a4c30666d04671f290c0103a8c773
Author(s)
Nguyen, Ngoc Cuong
Peraire, Jaime
Date Issued
January 2015
Journal
Computer Methods in Applied Mechanics and Engineering
Publisher
Elsevier
Citation
Nguyen, N.C., and J. Peraire. “Gaussian Functional Regression for Linear Partial Differential Equations.” Computer Methods in Applied Mechanics and Engineering 287 (2015): 69–89.
Version
Author's final manuscript
Abstract
In this paper, we present a new statistical approach to the problem of incorporating experimental observations into a mathematical model described by linear partial differential equations (PDEs) to improve the prediction of the state of a physical system. We augment the linear PDE with a functional that accounts for the uncertainty in the mathematical model and is modeled as a Gaussian process. This gives rise to a stochastic PDE which is characterized by the Gaussian functional. We develop a Gaussian functional regression method to determine the posterior mean and covariance of the Gaussian functional, thereby solving the stochastic PDE to obtain the posterior distribution for our prediction of the physical state. Our method has the following features which distinguish itself from other regression methods. First, it incorporates both the mathematical model and the observations into the regression procedure. Second, it can handle the observations given in the form of linear functionals of the field variable. Third, the method is non-parametric in the sense that it provides a systematic way to optimally determine the prior covariance operator of the Gaussian functional based on the observations. Fourth, it provides the posterior distribution quantifying the magnitude of uncertainty in our prediction of the physical state. We present numerical results to illustrate these features of the method and compare its performance to that of the standard Gaussian process regression.
MIT Department
Massachusetts Institute of Technology. Department of Earth, Atmospheric, and Planetary Sciences
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
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DOI of Published Version
https://doi.org/10.1016/j.cma.2015.01.008
