Adjoint shadowing directions in hyperbolic systems for sensitivity analysis

Abstract

For hyperbolic diffeomorphisms, we define adjoint shadowing directions as a bounded inhomogeneous adjoint solution whose initial condition has zero component in the unstable adjoint direction. For hyperbolic flows, we define adjoint shadowing directions similarly, with the additional requirement that the average of its inner-product with the trajectory direction is zero. In both cases, we show unique existence of adjoint shadowing directions, and how they can be used for adjoint sensitivity analysis. Our work set a theoretical foundation for efficient adjoint sensitivity methods for long-time-averaged objectives such as NILSAS.

22 pages, 1 figure. The algorithm for the adjoint shadowing direction is at arXiv:1801.08674

References in corpus (2)

• Adjoint sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Adjoint Shadowing (NILSAS)
• Sensitivity analysis on chaotic dynamical systems by Finite Difference Non-Intrusive Least Squares Shadowing (FD-NILSS)

Cited by in corpus (6)

• Hyperbolicity, shadowing directions and sensitivity analysis of a turbulent three-dimensional flow
• Adjoint sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Adjoint Shadowing (NILSAS)
• Sensitivity analysis on chaotic dynamical systems by Finite Difference Non-Intrusive Least Squares Shadowing (FD-NILSS)
• Variational optimization and data assimilation in chaotic time-delayed systems with automatic-differentiated shadowing sensitivity
• Approximating linear response by nonintrusive shadowing algorithms
• Fast differentiation of hyperbolic chaos