Global Optimisation of a Goal-Oriented Reduced-Order Model

Abstract

Goal-oriented reduced-order models (GOROM) have the potential to represent complex dynamics in an interpretable way. In the GOROM procedure, modes are found which when used in a Galerkin projection of the governing equations, return the desired goal function with minimum error. Determining such models leads to a high-dimensional optimisation problem, which may have multiple extrema, making purely local optimisation inapplicable.
In this thesis, a global optimisation algorithm was derived for GOROM. The optimisation response surface was approximated with a surrogate model, built using the Kriging technique. In each iteration, the surrogate model was improved by including the objective function of the three points with the highest expected improvements. This iterative process was stopped when the expected improvement fell below a threshold.
First, the performance of the algorithm was confirmed for a simple non-GOROM optimisation. Then the algorithm was applied to GOROM optimisation for the forced Burgers equation, initially using a complex forcing function giving multiple minima and later to a forcing function derived from a DNS of a turbulent channel flow. The initial forcing required on the order of tens of degrees of freedom to be optimised. The corresponding response surface proved similar to that of the simple problems, showing these optimisations to be tractable.
For the DNS forcing, hundreds of coefficients needed to be optimised. Using the developed algorithm a better optimal was found than with the local algorithm, although substantially more computational resources would be required to confirm global optimality.