Sensitivity analysis (SA) of time averaged quantities due to parameter change and
uncertainty quantification (UQ) due to stochastic parameter variation are important
analysis tools for characterisation and optimisation of complex engineering systems.
Commonly used methods for SA (for example the standard adjoint method) fail to
produce accurate estimates when the dynamical system displays chaotic behaviour.
For UQ, existing methods (such as the Polynomial Chaos expansion) are very com-
putationally demanding when the stochastic input is high dimensional. The main
purpose of this thesis is to provide a framework for the efficient SA and UQ analysis
of chaotic systems.
In the field of SA, a novel framework for computing the gradients of time-averaged
quantities to changes in system parameters is presented, referred to as the shadowing
harmonic method. The algorithm is based on the shadowing lemma and solves the
sensitivity problem in the frequency domain. An iterative method is proposed that
can reduce the memory and computing requirements and allow for extension to
large-scale dynamical systems.
In the field of UQ, a method based on the generalized Polynomial Chaos (gPC) is
presented that can produce accurate statistics for a Quantity of Interest (QoI) in
the presence of a large number of stochastic variables, at a greatly reduced com-
putational cost. In its simplest formulation, the cost of the method is independent
from the number of stochastic parameters. The method uses sensitivity information,
obtained efficiently from the adjoint SA approach, to enrich the least squares system
derived to compute the coefficients of the gPC. This approach is found to produce
accurate sensitivities at a cost that is reduced by a factor equal to the dimension
of the input stochastic space. The method is tested on chaotic and non-chaotic
systems.
Finally, the two methods are combined to produce a framework for the sensitivity
analysis and uncertainty quantification of chaotic systems. The framework can be
directly applied to large scale problems, such as turbulent flows.