  In many time-dependent problems of practical interest the parameters and/or
initial conditions entering the equations describing the evolution of the
various quantities exhibit uncertainty. One way to address the problem of how
this uncertainty impacts the solution is to expand the solution using
polynomial chaos expansions and obtain a system of differential equations for
the evolution of the expansion coefficients. We present an application of the
Mori-Zwanzig (MZ) formalism to the problem of constructing reduced models of
such systems of differential equations. In particular, we construct reduced
models for a subset of the polynomial chaos expansion coefficients that are
needed for a full description of the uncertainty caused by uncertain parameters
or initial conditions.
  Even though the MZ formalism is exact, its straightforward application to the
problem of constructing reduced models for estimating uncertainty involves the
computation of memory terms whose cost can become prohibitively expensive. For
those cases, we present a Markovian reformulation of the MZ formalism which can
lead to approximations that can alleviate some of the computational expense
while retaining an accuracy advantage over reduced models that discard the
memory altogether. Our results support the conclusion that successful reduced
models need to include memory effects.
