  We formulate a new projection-based reduced-ordered modeling technique for
non-linear dynamical systems. The proposed technique, which we refer to as the
Adjoint Petrov-Galerkin (APG) method, is derived by decomposing the generalized
coordinates of a dynamical system into a resolved coarse-scale set and an
unresolved fine-scale set. A Markovian finite memory assumption within the
Mori-Zwanzig formalism is then used to develop a reduced-order representation
of the coarse-scales. This procedure leads to a closed reduced-order model that
displays commonalities with the adjoint stabilization method used in finite
elements. The formulation is shown to be equivalent to a Petrov-Galerkin method
with a non-linear, time-varying test basis, thus sharing some similarities with
the least-squares Petrov-Galerkin method. Theoretical analysis examining a
priori error bounds and computational cost is presented. Numerical experiments
on the compressible Navier-Stokes equations demonstrate that the proposed
method can lead to improvements in numerical accuracy, robustness, and
computational efficiency over the Galerkin method on problems of practical
interest. Improvements in numerical accuracy and computational efficiency over
the least-squares Petrov-Galerkin method are observed in most cases.
