[Paper Review] Approximating linear response by nonintrusive shadowing algorithms

Angxiu Ni|arXiv (Cornell University)|Mar 22, 2020

Theoretical and Computational Physics40 references4 citations

TL;DR

This paper establishes that nonintrusive shadowing algorithms compute only the 'shadowing contribution' to the linear response in chaotic systems, not the full derivative. It proves that the remaining 'unstable contribution' is the systematic error of shadowing methods, and shows this error is small when unstable dimensions are low, validating shadowing's success in high-dimensional systems like fluid dynamics.

ABSTRACT

Nonintrusive shadowing algorithms efficiently compute $v$, the difference between shadowing trajectories, then use $v$ to compute derivatives of averaged objectives of chaos with respect to parameters of the dynamical system. However, previous proofs of shadowing methods wrongly assume that shadowing trajectories are representative. In contrast, the linear response formula is proved rigorously, but is more difficult to compute. We prove that $v$ gives only a part, called the shadowing contribution, of the linear response; hence, the other part, the unstable contribution, is the systematic error of shadowing methods. For systems with a small ratio of unstable dimensions, with some further statistical assumptions, we show that the unstable contribution is small. We also briefly describe an algorithm for the unstable contribution, which is simpler to derive but less efficient than the fast linear response algorithm. Moreover, we prove the convergence of the nonintrusive shadowing algorithm, the fastest shadowing algorithm, to $v$ and to the shadowing contribution.

Motivation & Objective

• To rigorously analyze the error in nonintrusive shadowing algorithms when approximating linear response in chaotic systems.
• To identify the systematic error in shadowing methods as the 'unstable contribution'—a missing part of the full linear response.
• To prove that the nonintrusive shadowing algorithm converges to the shadowing contribution, not the full linear response.
• To provide conditions under which the unstable contribution is small, justifying shadowing's empirical success in high-dimensional systems.
• To lay the theoretical groundwork for a fast linear response algorithm by decomposing the response into shadowing and unstable contributions.

Proposed method

• Uses a decomposition of the linear response into two parts: the shadowing contribution (computed by shadowing) and the unstable contribution (the error).
• Applies the nonintrusive shadowing algorithm, which computes the shadowing direction $ v $ using only $ m $ solutions of the tangent equation, where $ m $ is the unstable dimension.
• Employs a sampling-based error analysis for the shadowing contribution, bounding the sampling error via correlation decay assumptions.
• Introduces Assumption 3 on exponential decay of correlations for $ \Phi_u v $, enabling a quantitative bound on the sampling error $ \|\tilde{e}^S\| $.
• Derives convergence bounds for the nonintrusive shadowing algorithm using Holder continuity and decay of correlations in uniform hyperbolic systems.
• Proposes a fast linear response algorithm in a follow-up work, using second-order tangent equations to compute the unstable contribution efficiently.

Experimental results

Research questions

• RQ1 What part of the linear response does the nonintrusive shadowing algorithm actually compute?
• RQ2 Why do shadowing methods succeed in high-dimensional chaotic systems like fluid dynamics despite known theoretical limitations?
• RQ3 What is the systematic error in shadowing methods, and under what conditions is it small?
• RQ4 Can the convergence of the nonintrusive shadowing algorithm to the shadowing contribution be rigorously proven?
• RQ5 How can the full linear response be recovered by combining shadowing and an efficient computation of the unstable contribution?

Key findings

• The nonintrusive shadowing algorithm computes only the 'shadowing contribution' to the linear response, not the full derivative.
• The systematic error of shadowing methods is the 'unstable contribution', which is the missing part of the linear response.
• For systems with low unstable dimension and under exponential correlation decay, the unstable contribution is small, explaining shadowing's empirical success.
• The sampling error in the shadowing contribution is bounded by $ \|\tilde{e}^S\| \leq \sqrt{\frac{2C_3}{K(1 - \kappa_3)}} \|v^A\| $, which decays as $ K^{-0.5} $ under Assumption 3.
• The nonintrusive shadowing algorithm converges to the shadowing contribution at the same rate as previous shadowing methods, confirming its accuracy for $ v $.
• The total error in approximating the linear response is bounded by the sum of the unstable contribution, sampling error, and numerical error, with the latter two vanishing as $ K \to \infty $.