PHYSICAL REVIEW E 101, 022223 (2020) Uncertainty quantification of sensitivities of time-average quantities in chaotic systems Kyriakos D. Kantarakias , * Karim Shawki, † and George Papadakis ‡ Department of Aeronautics, Imperial College, London SW7 2AZ, United Kingdom (Received 21 November 2019; accepted 10 February 2020; published 28 February 2020) We consider time-average quantities of chaotic systems and their sensitivity to system parameters. When the parameters are random variables with a prescribed probability density function, the sensitivities are also random. The central aim of the paper is to study and quantify the uncertainty of the sensitivities; this is useful to know in robust design applications. To this end, we couple the nonintrusive polynomial chaos expansion (PCE) with the multiple shooting shadowing (MSS) method, and apply the coupled method to two standard chaotic systems, the Lorenz system and the Kuramoto-Sivashinsky equation. The method leads to accurate results that match well with Monte Carlo simulations (even for low chaos orders, at least for the two systems examined), but it is costly. However, if we apply the concept of shadowing to the system trajectories evaluated at the quadrature integration points of PCE, then the resulting regularization can lead to significant computational savings. We call the new method shadowed PCE (sPCE). DOI: 10.1103/PhysRevE.101.022223 I. INTRODUCTION All practical systems exhibit a degree of uncertainty in the values of the system parameters. For example, the an- gle of attack or the geometric shape of an airfoil exhibit a degree of randomness (due to free-stream turbulence or surface roughness, respectively). Uncertainty quantification (UQ) methods aim at modeling the effect of uncertainties in a computationally efficient manner. They define a quantity of interest (QoI), for example, lift or drag of an airfoil, and a set of uncertain parameters (such as angle of attack, airfoil shape, etc.) that affect the QoI. The objective of UQ methods is to evaluate the statistical behavior of the QoI using the available probabilistic information of the uncertain parameters and the input-output relation of the system. In this paper, we consider systems in which the input-output relation is governed by a set of nonlinear partial differential equations (PDEs). For a review of the available UQ methods with a particular focus on Computational fluid dynamics applications (where the governing PDEs are the Navier-Stokes equations), the reader is referred to Refs. [1,2]. The most commonly used UQ method is the polynomial chaos expansion (PCE), due to its computational efficiency for a small number of uncertain parameters. The mathematical framework was developed by Wiener [3] for Hermitte polyno- mials and was later generalized in Ref. [4] for every polyno- mial in the Wiener-Askey scheme. The method uses a spectral representation of the uncertain quantities in an orthonormal stochastic space and computes the spectral coefficients with Galerkin projection. There are two approaches for applying PCE to dynamical systems, the intrusive and nonintrusive; below they will are * k.kantarakias@imperial.ac.uk † karim.shawki14@imperial.ac.uk ‡ Corresponding author: g.papadakis@imperial.ac.uk denoted as iPC and niPC, respectively. In niPC, the QoI is written in spectral form and the unknown expansion coef- ficients are computed by evaluating the Galerkin integral at specific integration points (or nodes). Each evaluation requires the integration of the PDE system for a particular set of values of the input random variables. In that sense, niPC is a black- box approach, i.e., all that is required is a code that provides the solution of the PDE system, which is called repeatedly for a set of specific inputs determined by the PCE algorithm. However, in the iPC [5], the system degrees of freedom are written in spectral form and a new set of equations, the iPC equations, that describe the evolution of the expansion coef- ficients, are extracted using Galerkin projection. This results in a coupled system of equations that must be integrated, usually with methods similar to the ones used for the original dynamical system. In steady systems, i.e., ones that do not exhibit time- dependent behavior, iPC is a well-established method with numerous applications in compressible and incompressible flows, reacting flows, flows in media, etc. (refer to Refs. [6–8] for a small sample of applications). Application of UQ to unsteady systems is much more in- volved. First, the iPC method was found to exhibit significant limitations. More specifically, as the unsteady system evolves from an initial state, the number of expansion polynomials must increase to maintain the accuracy of the expansion [9–14]. The systems do not necessarily have to exhibit chaotic behavior for this problem to appear; the root of the problem lies on the nonlinear interactions that are present in either chaotic or nonchaotic systems. This is aptly demonstrated in Ref. [9] using a linear decay equation with a uniformly distributed random decay coefficient (the limits of the distri- bution are such that the latter always remain positive). Since both the coefficient and the solution are random, a quadratic nonlinearity appears, even though the deterministic equation is linear. As the system evolves, the solution starts to develop its own random characteristics, that deviate from those close 2470-0045/2020/101(2)/022223(10) 022223-1 ©2020 American Physical Society