  In this work, we consider the problem of optimal design of an acoustic cloak
under uncertainty and develop scalable approximation and optimization methods
to solve this problem. The design variable is taken as an infinite-dimensional
spatially-varying field that represents the material property, while an
additive infinite-dimensional random field represents the variability of the
material property or the manufacturing error. Discretization of this optimal
design problem results in high-dimensional design variables and uncertain
parameters. To solve this problem, we develop a computational approach based on
a Taylor approximation and an approximate Newton method for optimization, which
is based on a Hessian derived at the mean of the random field. We show our
approach is scalable with respect to the dimension of both the design variables
and uncertain parameters, in the sense that the necessary number of acoustic
wave propagations is essentially independent of these dimensions, for numerical
experiments with up to one million design variables and half a million
uncertain parameters. We demonstrate that, using our computational approach, an
optimal design of the acoustic cloak that is robust to material uncertainty is
achieved in a tractable manner. The optimal design under uncertainty problem is
posed and solved for the classical circular obstacle surrounded by a
ring-shaped cloaking region, subjected to both a single-direction
single-frequency incident wave and multiple-direction multiple-frequency
incident waves. Finally, we apply the method to a deterministic large-scale
optimal cloaking problem with complex geometry, to demonstrate that the
approximate Newton method's Hessian computation is viable for large, complex
problems.
