A Trajectory-Driven Algorithm for Differentiating SRB Measures on Unstable Manifolds

Journal Article · 08 February 2022 · SIAM Journal on Scientific Computing

DOI: https://doi.org/10.1137/21m1431916 · OSTI ID:1844554

Śliwiak, Adam A.  ^[1]; Wang, Qiqi ^[2]

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1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States); Massachusetts Institute of Technology
2. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

Sinai-Ruelle-Bowen (SRB) measures are limiting stationary distributions describing the statistical behavior of chaotic dynamical systems. Directional derivatives of SRB measure densities conditioned on unstable manifolds are critical in the sensitivity analysis of hyperbolic chaos. These derivatives, known as the SRB density gradients, are by-products of the regularization of Lebesgue integrals appearing in the original linear response expression. In this paper, we propose a novel trajectory- driven algorithm for computing the SRB density gradient defined for systems with high-dimensional unstable manifolds. We apply the concept of measure preservation together with the chain rule on smooth manifolds. Due to the recursive one-step nature of our derivations, the proposed procedure is memory-efficient and can be naturally integrated with existing Monte Carlo schemes widely used in computational chaotic dynamics. Here, we numerically show the exponential convergence of our scheme, analyze the computational cost, and present its use in the context of Monte Carlo integration.

• Research Organization: Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: NA0003993
• OSTI ID: 1844554
• Journal Information: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 1 Vol. 44; ISSN 1064-8275
• Publisher: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement
• Country of Publication: United States
• Language: English