Multiple Shooting Shadowing for Sensitivity Analysis of Chaotic Systems and Turbulent fluid flows

Abstract

Sensitivity analysis methods are important tools for research and design with simulations. Many important simulations exhibit chaotic dynamics, including scale-resolving turbulent fluid flow simulations. Unfortunately , conventional sensitivity analysis methods are unable to compute useful gradient information for long-time-averaged quantities in chaotic dynamical systems. Sensitivity analysis with least squares shadowing (LSS) can compute useful gradient information for a number of chaotic systems, including simulations of chaotic vortex shedding and homogeneous isotropic turbulence. However, this gradient information comes at a very high computational cost. This paper presents multiple shooting shadowing (MSS), a more com-putationally efficient shadowing approach than the original LSS approach. Through an analysis of the convergence rate of MSS, it is shown that MSS can have lower memory usage and run time than LSS.

Application of Generalized Polynomial Chaos for Quantification of Uncertainties of Time Averages and Their Sensitivities in Chaotic Systems

“…Equation (5b) enforces the continuity of v across consecutive segments, Equation (5c) is the linearized version of Equation (1) around s, and Equation (5d) enforces the direction v to be normal to f. Finally, η is a time-dilation term between the reference and the shadowing trajectory. We briefly describe the solution of the minimization problem in Equation 5, and more details can be found in Reference [21,22]. We reformulate Equation 5as…”

“…Although computationally expensive, the method can give values for the sensitivities of time-averaged quantities that match well with finite differences. In this work, the preconditioned Multiple-Shooting (MSS) [21,22], a computationally more efficient version of the LSS, is used to evaluate sensitivities of time-averages.…”

Lyapunov spectrum of scale-resolving turbulent simulations. Application to chaotic adjoints

“…In particular, the cost of the two main adjoint-based sensitivity methods in the literature, namely the Least Squares Shadowing (LSS) 26 and the Ensemble Adjoint (EA) methods, increases with the number and/or magnitude of positive Lyapunov exponents in the discrete system. The computational cost of shadowing-based approaches, including the original LSS method, the Multiple Shooting Shadowing (MSS) method 2 and the Non-Intrusive LSS (NILSS) method, 3,17 is proportional to the number of positive Lyapunov exponents. Under optimistic assumptions, including uniform hyperbolicity and exponential decay of correlations, Chandramoorthy et al 5,6 showed that the mean squared error in the EA estimator is a power law of the computational cost, where the power constant depends on the leading exponent.…”