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Adjoint of Least Squares Shadowing: Existence, Uniqueness and Coarse Domain Discretization

Pranshul Thakur, Siva Nadarajah

TL;DR

This work addresses sensitivity analysis in chaotic dynamical systems by focusing on the adjoint of the least-squares shadowing (LSS) framework. It proves existence and uniqueness of the continuous adjoint LSS solution and derives a sharp, horizon-independent bound on the LSS conditioning, enabling a robust convergence result: the adjoint LSS sensitivity converges to the true sensitivity at a rate of $\mathcal{O}\left(\frac{1}{\sqrt{T}}\right)$ as the integration time $T$ grows. The analysis connects the conditioning to the time-dilation factor $\alpha^2$ and shows that adjoint boundary conditions do not alter the convergence rate, while discretization errors scale with the local truncation error. A key practical implication is that the adjoint LSS can be discretized on a coarser time grid than the primal, reducing the space-time solve cost without sacrificing asymptotic accuracy. Numerical tests on Lorenz 63 and a coupled oscillator corroborate the theory, demonstrating bounded conditioning and correct convergence behavior for both homogeneous and non-homogeneous adjoint BCs and illustrating the benefits of coarse-time adjoint discretization.

Abstract

Chaotic dynamical systems are characterized by the sensitive dependence of trajectories on initial conditions. Conventional sensitivity analysis of time-averaged functionals yields unbounded sensitivities when the simulation is chaotic. The least squares shadowing (LSS) is a popular approach to computing bounded sensitivities in the presence of chaotic dynamical systems. The current paper proves the existence, uniqueness, and boundedness of the adjoint of the LSS equations. In particular, the analysis yields a sharper bound on the condition number of the LSS equations than currently demonstrated in existing literature and shows that the condition number is bounded for large integration times. The derived bound on condition number also shows a relation between the conditioning of the LSS and the time dilation factor which is consistent with the trend numerically observed in the previous LSS literature. Furthermore, using the boundedness of the condition number for large integration times, we provide an alternate proof to (Chater et al., 2017) of the convergence of the LSS sensitivity to the true sensitivity at the rate of $\mathcal{O}\left(\frac{1}{\sqrt{T}}\right)$ regardless of the boundary conditions imposed on the adjoint, as long as the adjoint boundary conditions are bounded. Existence and uniqueness of the solution to the continuous-in-time adjoint LSS equation ensure that the LSS equation can be discretized independently of the primal equation and that the true LSS adjoint solution is recovered as the time step is refined. This allows for the adjoint LSS equation to be discretized on a coarser time domain than that of the primal governing equation to reduce the cost of solving the linear space-time system.

Adjoint of Least Squares Shadowing: Existence, Uniqueness and Coarse Domain Discretization

TL;DR

This work addresses sensitivity analysis in chaotic dynamical systems by focusing on the adjoint of the least-squares shadowing (LSS) framework. It proves existence and uniqueness of the continuous adjoint LSS solution and derives a sharp, horizon-independent bound on the LSS conditioning, enabling a robust convergence result: the adjoint LSS sensitivity converges to the true sensitivity at a rate of as the integration time grows. The analysis connects the conditioning to the time-dilation factor and shows that adjoint boundary conditions do not alter the convergence rate, while discretization errors scale with the local truncation error. A key practical implication is that the adjoint LSS can be discretized on a coarser time grid than the primal, reducing the space-time solve cost without sacrificing asymptotic accuracy. Numerical tests on Lorenz 63 and a coupled oscillator corroborate the theory, demonstrating bounded conditioning and correct convergence behavior for both homogeneous and non-homogeneous adjoint BCs and illustrating the benefits of coarse-time adjoint discretization.

Abstract

Chaotic dynamical systems are characterized by the sensitive dependence of trajectories on initial conditions. Conventional sensitivity analysis of time-averaged functionals yields unbounded sensitivities when the simulation is chaotic. The least squares shadowing (LSS) is a popular approach to computing bounded sensitivities in the presence of chaotic dynamical systems. The current paper proves the existence, uniqueness, and boundedness of the adjoint of the LSS equations. In particular, the analysis yields a sharper bound on the condition number of the LSS equations than currently demonstrated in existing literature and shows that the condition number is bounded for large integration times. The derived bound on condition number also shows a relation between the conditioning of the LSS and the time dilation factor which is consistent with the trend numerically observed in the previous LSS literature. Furthermore, using the boundedness of the condition number for large integration times, we provide an alternate proof to (Chater et al., 2017) of the convergence of the LSS sensitivity to the true sensitivity at the rate of regardless of the boundary conditions imposed on the adjoint, as long as the adjoint boundary conditions are bounded. Existence and uniqueness of the solution to the continuous-in-time adjoint LSS equation ensure that the LSS equation can be discretized independently of the primal equation and that the true LSS adjoint solution is recovered as the time step is refined. This allows for the adjoint LSS equation to be discretized on a coarser time domain than that of the primal governing equation to reduce the cost of solving the linear space-time system.

Paper Structure

This paper contains 11 sections, 11 theorems, 83 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Least squares shadowing approach to sensitivity analysis
  4. Adjoint of least squares shadowing
  5. Adjoint shadowing direction
  6. Convergence of the adjoint LSS sensitivity
  7. Discussion
  8. Numerical test cases
  9. Lorenz 63
  10. Coupled oscillator
  11. Conclusion

Key Result

Lemma 5.1

\newlabellm:adjoint_dot_f0 The shadowing direction, $\psi^\infty$, satisfying eq:conventional_adjoint and eq:avg_adjoint also satisfies

Figures (8)

  • Figure 1: Lorenz attractor
  • Figure 2: Variation of the LSS condition number, $\|\mathcal{L}^{-1}\|$, with time.
  • Figure 3: Convergence of sensitivity with (a) homogeneous and (b) non-homogeneous adjoint boundary conditions.
  • Figure 4: Convergence of sensitivity with grid size using adjoint LSS.
  • Figure 5: Attractor of the coupled oscillator
  • ...and 3 more figures

Theorems & Definitions (27)

  • Lemma 5.1
  • Proof 1
  • Lemma 6.1
  • Proof 2
  • Definition 6.2
  • Lemma 6.3
  • Proof 3
  • Lemma 6.4
  • Proof 4
  • Lemma 6.5
  • ...and 17 more