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

Abstract

Abstract Ruelle (1997 Commun. Math. Phys. 187 227–41; 2003 Commun. Math. Phys. 234 185–90) (see also Jiang 2012 Ergod. Theor. Dynam. Syst. 32 1350–69) 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 (Chandramoorthy and Wang 2022 SIAM J. Appl. Dyn. Syst. 21 735–81). 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.