Preliminary Investigation of Hessian Approximation with an Orthogonal Modal Parameterisation for Accelerated Shape Optimisation

Newton's method for optimisation presents an attractive quadratic convergence rate, however practical application to engineering optimisation problems is hampered by the computational cost and complexity associated with obtaining the required second-order information.
In this paper, preliminary investigations are presented for approximating incomplete Hessians for via a compact orthogonal shape parameterisation.
Orthogonal modes have been shown to be very compact at representing design spaces for aerodynamic applications which motivates the current work on the use of orthogonal shape modes to inexpensively sample key curvature information via a truncated Hessian in order to accelerate optimisation convergence.
A subset of orthogonal modes is used to construct a partial Hessian which is applied to precondition the shape optimisation problem equivalently to an exactly initialised quasi-Newton method.
The methodology is tested on a simple inviscid pressure matching problem and it is shown to be effective at improving optimisation convergence rate and hence reducing computational cost.
Sampling with the higher-frequency orthogonal shape modes gives a good approximation to the largest eigenvalues of the Hessian, and a simple modification to the partial eigenvalue set is shown to further improve optimisation convergence rate.