  Optimal prediction methods compensate for a lack of resolution in the
numerical solution of time-dependent differential equations through the use of
prior statistical information. We present a new derivation of the basic
methodology, show that field-theoretical perturbation theory provides a useful
device for dealing with quasi-linear problems, and provide a nonlinear example
that illuminates the difference between a pseudo-spectral method and an optimal
prediction method with Fourier kernels. Along the way, we explain the
differences and similarities between optimal prediction, the representer method
in data assimilation, and duality methods for finding weak solutions. We also
discuss the conditions under which a simple implementation of the optimal
prediction method can be expected to perform well.
