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We develop the NILSAS algorithm, which performs adjoint sensitivity analysis of chaotic systems via computing the adjoint shadowing direction. NILSAS constrains its minimization to the adjoint unstable subspace, and can be implemented with little modification to existing adjoint solvers. The computational cost of NILSAS is independent of the number of parameters. We demonstrate NILSAS on the Lorenz 63 system and a turbulent three-dimensional flow over a cylinder. • NILSAS performs adjoint sensitivity analysis for chaotic systems via computing adjoint shadowing directions. • Computational cost of NILSAS is independent of the number of system parameters. • NILSAS constrains minimization to the unstable adjoint subspace. • NILSAS can be implemented with little modifications to existing adjoint solvers. • We demonstrate NILSAS on a weakly turbulent three dimensional flow over a cylinder. [ABSTRACT FROM AUTHOR]

• We develop the FD-NILSS algorithm for computing sensitivities of long-time averaged objetives in chaotic systems. • FD-NILSS requires little modification to existing simulation software, and it does not require linearized solvers. • FD-NILSS computes correct sensitivities of a chaotic flow past a 3-D cylinder, with cost similar to numerical simulation. • We give a formula for solving the least-squares problem, which can be applied in both FD-NILSS and NILSS. We present the Finite Difference Non-Intrusive Least Squares Shadowing (FD-NILSS) algorithm for computing sensitivities of long-time averaged quantities in chaotic dynamical systems. FD-NILSS does not require tangent solvers, and can be implemented with little modification to existing numerical simulation software. We also give a formula for solving the least-squares problem in FD-NILSS, which can be applied in NILSS as well. Finally, we apply FD-NILSS for sensitivity analysis of a chaotic flow over a 3-D cylinder at Reynolds number 525, where FD-NILSS computes accurate sensitivities and the computational cost is in the same order as the numerical simulation. [ABSTRACT FROM AUTHOR]

For a parameterized hyperbolic system du/dt=f(u,s) the derivative of the ergodic average 〈J〉 = limT→∞1/T ∫ T0 J(u(t); s) to the parameter s can be computed via the Least Squares Shadowing algorithm (LSS). We assume that the sytem is ergodic which means that 〈J〉 depends only on s (not on the initial condition of the hyperbolic system). After discretizing this continuous system using a fixed timestep, the algorithm solves a constrained least squares problem and, from the solution to this problem, computes the desired derivative d\〈J〉/ds. The purpose of this paper is to prove that the value given by the LSS algorithm approaches the exact derivative when the timespan used to formulate the least squares problem grows to infinity. It then illustrates the convergence result through a numerical example. [ABSTRACT FROM AUTHOR]

This paper develops the Non-Intrusive Least Squares Shadowing (NILSS) method, which computes the sensitivity for long-time averaged objectives in chaotic dynamical systems. In NILSS, we represent a tangent solution by a linear combination of one inhomogeneous tangent solution and several homogeneous tangent solutions. Next, we solve a least squares problem using this representation; thus, the resulting solution can be used for computing sensitivities. NILSS is easy to implement with existing solvers. In addition, for chaotic systems with many degrees of freedom but few unstable modes, NILSS has a low computational cost. NILSS is applied to two chaotic PDE systems: the Lorenz 63 system and a CFD simulation of flow over a backward-facing step. In both cases, the sensitivities computed by NILSS reflect the trends in the long-time averaged objectives of dynamical systems. [ABSTRACT FROM AUTHOR]

This paper develops a variant of the Least Squares Shadowing (LSS) method, which has successfully computed the derivative for several chaotic ODEs and PDEs. The development in this paper aims to simplify Least Squares Shadowing method by improving how time dilation is treated. Instead of adding an explicit time dilation term as in the original method, the new variant uses windowing, which can be more efficient and simpler to implement, especially for PDEs. [ABSTRACT FROM AUTHOR]