  In this paper, practically computable low-order approximations of potentially
high-dimensional differential equations driven by geometric rough paths are
proposed and investigated. In particular, equations are studied that cover the
linear setting, but we allow for a certain type of dissipative nonlinearity in
the drift as well. In a first step, a linear subspace is found that contains
the solution space of the underlying rough differential equation (RDE). This
subspace is associated to covariances of linear Ito-stochastic differential
equations which is shown exploiting a Gronwall lemma for matrix differential
equations. Orthogonal projections onto the identified subspace lead to a first
exact reduced order system. Secondly, a linear map of the RDE solution
(quantity of interest) is analyzed in terms of redundant information meaning
that state variables are found that do not contribute to the quantity of
interest. Once more, a link to Ito-stochastic differential equations is used.
Removing such unnecessary information from the RDE provides a further dimension
reduction without causing an error. Finally, we discretize a linear parabolic
rough partial differential equation in space. The resulting large-order RDE is
subsequently tackled with the exact reduction techniques studied in this paper.
We illustrate the enormous complexity reduction potential in the corresponding
numerical experiments.
