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 longtime-averaged objectives such as NILSAS.

Fast linear response algorithm for differentiating chaos

“…Together with the so called 'little-intrusive' technique for efficiently computing projection operators [22,24], we now have enough tools to compute the unstable contribution, hence the linear response formula, by the expressions in theorem 1. However, that would cause several numerical issues, which are explained and resolved in the next section.…”

“…It was also applied to thermoacoustic systems [17]. The adjoint shadowing lemma and corresponding non-intrusive algorithms were also devised [22,25,6]. A side result of adjoint shadowing lemma is an efficient algorithm, the 'little-intrusive' formulation for computing the oblique projection operators, which uses only the unstable tangent and adjoint subspace, but not the full stable subspace: this technique can be carried to other algorithms to improve efficiency.…”

Adjoint sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Adjoint Shadowing (NILSAS)

“…In numerical implementations, we typically solve adjoint equations backward in time. This is because, as shown in [30], when solving backward in time, the dimension of the unstable adjoint subspace is the same as the unstable tangent subspace, which is typically much lower than m. On the other hand, if we solve the adjoint equation forward in time, the unstable subspace has much higher dimension, causing strong numerical instability. Definition 2.…”

“…We assume our system is uniform hyperbolic and it has a bounded global attractor. Definition of hyperbolicity can be found in most textbook on dynamical system such as [32], and readers may also refer to [30] for a definition using the same notation as this paper. Uniform hyperbolicity requires that the tangent space can be split into stable subspace, unstable subspace, and a neutral subspace of dimension one.…”

“…Recently, we defined the adjoint shadowing direction for both hyperbolic flows and diffeomorphisms, which is a bounded inhomogeneous adjoint solution with several other properties [30]. We showed that the adjoint shadowing direction exists uniquely on a given trajectory, and can be used for adjoint sensitivity analysis.…”