  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 {\em
Gaussian process}. This gives rise to a stochastic PDE which is characterized
by the Gaussian functional. We develop a {\em functional Gaussian process
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.
