Exact dimension reduction for rough differential equations

TL;DR

This paper develops exact dimension reduction techniques for high-dimensional rough differential equations, enabling efficient computation by identifying and removing redundant information and reducing the problem size significantly.

Contribution

It introduces a novel method for exact dimension reduction of rough differential equations, combining covariance-based subspace identification and redundancy elimination.

Findings

Significant reduction in computational complexity demonstrated.

Exact reduced order systems preserve the original solution.

Applicable to high-dimensional rough PDEs with complex dynamics.

Abstract

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.