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.

翻译：本文提出并研究了由几何粗糙路径驱动的高维微分方程的实用可计算低阶近似。具体而言，我们研究的方程涵盖线性框架，但允许漂移项中存在特定类型的耗散非线性。首先，找到一个包含底层粗糙微分方程解空间的线性子空间，该子空间与线性伊藤随机微分方程的协方差相关联，这一关系通过矩阵微分方程的Gronwall引理得以证明。将正交投影应用于该子空间后，得到了首个精确降阶系统。其次，针对RDE解的线性映射（关注量），我们分析其中的冗余信息，即识别出不贡献于关注量的状态变量。这里再次利用与伊藤随机微分方程的联系。从RDE中移除这类冗余信息可在不引入误差的情况下进一步降维。最后，我们对线性抛物型粗糙偏微分方程进行空间离散化，随后采用本文研究的精确降维技术处理所得的高阶RDE。数值实验展示了该方法的巨大复杂度降低潜力。