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, e.g., 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.

在这项工作中，我们考虑了不确定性下的声学披风的优化设计问题，并开发了可扩展的近似和优化方法来解决该问题。设计变量被视为代表材料属性的无穷维空间变化字段，而加性无限维随机字段则代表例如材料属性的可变性或制造误差。最佳设计问题的离散化导致高维设计变量和不确定参数。为了解决此问题，我们开发了一种基于泰勒近似和近似牛顿优化方法的计算方法，该方法基于在随机场均值处得出的Hessian进行优化。我们展示了我们的方法相对于设计变量和不确定参数的尺寸而言是可扩展的，这意味着对于多达一百万个设计变量和一半的数值实验，声波传播的必要数量基本上与这些尺寸无关。一百万个不确定参数。我们证明，使用我们的计算方法，可以以易处理的方式实现对材料不确定性具有鲁棒性的声学披风的最佳设计。针对单向单频入射波和多向多频入射波同时受环形掩蔽区域包围的经典圆形障碍物，提出了不确定性问题下的最优设计。最后，