We study the numerical solution of forward and inverse time-harmonic acoustic scattering problems by randomly shaped obstacles in three-dimensional space using a fast isogeometric boundary element method. Within the isogeometric framework, realizations of the random scatterer can efficiently be computed by simply updating the NURBS mappings which represent the scatterer. This way, we end up with a random deformation field. In particular, we show that it suffices to know the deformation field’s expectation and covariance at the scatterer’s boundary to model the surface’s Karhunen–Loève expansion. Leveraging on the isogeometric framework, we employ multilevel quadrature methods to approximate quantities of interest such as the scattered wave’s expectation and variance. By computing the wave’s Cauchy data at an artificial, fixed interface enclosing the random obstacle, we can also directly infer quantities of interest in free space. Adopting the Bayesian paradigm, we finally compute the expected shape and variance of the scatterer from noisy measurements of the scattered wave at the artificial interface. Numerical results for the forward and inverse problems validate the proposed approach.

我们使用快速等几何边界元方法研究了三维空间中随机形状障碍物的正向和逆时谐声学散射问题的数值解。在等几何框架内，通过简单地更新表示散射体的 NURBS 映射，可以有效地计算随机散射体的实现。这样，我们最终会得到一个随机变形场。特别是，我们表明知道散射体边界处的变形场的期望和协方差就足以模拟表面的 Karhunen-Loève 膨胀。利用等几何框架，我们采用多级正交方法来近似感兴趣的数量，例如散射波的期望和方差。通过人工计算波的柯西数据，封闭随机障碍物的固定界面，我们也可以直接推断自由空间中感兴趣的数量。采用贝叶斯范式，我们最终根据人工界面处散射波的噪声测量来计算散射体的预期形状和方差。正向和逆向问题的数值结果验证了所提出的方法。