Leveraging the Linear Response Theory in Sensitivity Analysis of Chaotic Dynamical Systems and Turbulent Flows

Abstract

The linear response theory (LRT) provides a set of powerful mathematical tools for the analysis of system’s reactions to controllable perturbation. In applied sciences, LRT is particularly useful in approximating parametric derivatives of observables induced by a dynamical system. These derivatives, usually referred to as sensitivities, are critical components of optimization, control, numerical error estimation, risk assessment and other advanced computational methodologies. Efficient computation of sensitivities in the presence of chaos has been a major and still unresolved challenge in the field. While chaotic systems are prevalent in several fields of science and engineering, including turbulence and climate dynamics, conventional methods for sensitivity analysis are doomed to failure due to the butterfly effect. This inherent property of chaos means that any pair of infinitesimally close trajectories separates exponentially fast triggering serious numerical issues.

A new promising method, known as the space-split sensitivity (S3), addresses the adverse butterfly effect and has several appealing features. S3 directly stems from Ruelle’s closed-form linear response formula involving Lebesgue integrals of input-output time correlations. Its linearly separable structure combined with the chain rule on smooth manifolds enables the derivation of ergodic-averaging schemes for sensitivities that rigorously converge in uniformly hyperbolic systems. Thus, S3 can be viewed as an LRT-based Monte Carlo method that averages data collected through regularized tangent equations along a random orbit. Despite the recent theoretical advancements, S3 in its current form is applicable to systems with one-dimensional unstable manifolds, which makes it useless for real-world models.

In this thesis, we extend the concept of space-splitting to systems of arbitrary dimension, develop generic linear response algorithms for hyperbolic dynamical systems, and demonstrate their performance using common physical models. In particular, this work offers three major contributions to the field of nonlinear dynamics. First, we propose a novel algorithm for differentiating ergodic measures induced by chaotic systems. These quantities are integral components of the S3 method and arise from 3 the partial integration of Ruelle’s ill-conditioned expression. Our algorithm uses the concept of quantile functions to parameterize multi-dimensional unstable manifolds and computes the time evolution of measure gradients in a recursive manner. We also demonstrate that the measure gradients can be utilized as indicators of the differentiability of statistics, and might dramatically reduce the statistical-averaging error in the case of highly-oscillatory observables. Second, we blend the proposed manifold description, algorithm for measure gradients, and linear decomposition of the input perturbation, to derive a complete set of tangent equations for all by-products of the regularization process. We prove that all the recursive equations converge exponentially fast in uniformly hyperbolic systems, regardless of the choice of initial conditions. This result is used to assemble efficient one-step Monte Carlo algorithms applicable to high-dimensional discrete and continuous-time systems. Third, we argue that the effect of measure gradient could be negligible compared to the total linear response if the model is statistically homogeneous. Consequently, one could accurately approximate the sought-after sensitivity by evolving in time a single inhomogeneous tangent that is orthogonal to the unstable subspace everywhere along an orbit. This drastically reduces the computational complexity of the full algorithm.

Every major step of theoretical and algorithmic developments is corroborated by several numerical examples. They also highlight aspects of the underlying dynamical systems, e.g., ergodic measure distributions, Lyapunov spectra, spatiotemporal structures of tangent solutions, that are relevant in the context of sensitivity analysis. This thesis considers different classes of chaotic systems, including low-dimensional discrete systems (e.g., cusp map, baker’s map, multi-dimensional solenoid map), ordinary differential equations (Lorenz oscillators) and partial differential equations (Kuramoto-Sivashinsky and 3D Navier-Stokes system).