Rigorous justification for the space-split sensitivity algorithm to compute linear response in Anosov systems

TL;DR

This paper provides a rigorous mathematical proof for the convergence of the space-split sensitivity (S3) algorithm, enabling reliable computation of linear response in high-dimensional chaotic systems like Anosov diffeomorphisms.

Contribution

We establish the existence of the S3 decomposition and prove the convergence of the algorithm's computations, justifying its use for sensitivity analysis in chaotic dynamical systems.

Findings

Proved the existence of the S3 decomposition.

Established convergence of the S3 algorithm.

Validated the mathematical foundation for practical sensitivity computations.

Abstract

Ruelle gave a formula for linear response of transitive Anosov diffeomorphisms. Recently, practically computable realizations of Ruelle's formula have emerged that potentially enable sensitivity analysis of certain high-dimensional chaotic numerical simulations encountered in the applied sciences. In this paper, we provide full mathematical justification for the convergence of one such efficient computation, the space-split sensitivity, or S3, algorithm. In S3, Ruelle's formula is computed as a sum of two terms obtained by decomposing the perturbation vector field into a coboundary and a remainder that is parallel to the unstable direction. Such a decomposition results in a splitting of Ruelle's formula that is amenable to efficient computation. We prove the existence of the S3 decomposition and the convergence of the computations of both resulting components of Ruelle's formula.