An Adaptive Algorithm for Rough Differential Equations

TL;DR

This paper introduces an adaptive algorithm for solving rough differential equations efficiently by balancing error control and computational cost, utilizing the log-ODE method and an error representation formula.

Contribution

The paper proposes a novel adaptive algorithm that dynamically chooses between grid refinement and order elevation for solving RDEs using the log-ODE method.

Findings

The adaptive algorithm effectively reduces computational cost while maintaining accuracy.

Numerical examples demonstrate improved efficiency over non-adaptive methods.

The error representation formula accurately guides the adaptive process.

Abstract

We present an adaptive algorithm for effectively solving rough differential equations (RDEs) using the log-ODE method. The algorithm is based on an error representation formula that accurately describes the contribution of local errors to the global error. By incorporating a cost model, our algorithm efficiently determines whether to refine the time grid or increase the order of the log-ODE method. In addition, we provide several examples that demonstrate the effectiveness of our adaptive algorithm in solving RDEs.