Low-Dimensional Spatial Embedding Method for Shape Uncertainty Quantification in Acoustic Scattering

This paper introduces a novel boundary integral approach of shape uncertainty quantification for the Helmholtz scattering problem in the framework of the so-called parametric method. The key idea is to form a low-dimensional spatial embedding within the family of uncertain boundary deformations via the Coarea formula. The embedding, essentially, encompasses any irregular behavior of the boundary deformations and facilitates a low-dimensional integration rule capturing the bulk variation of output functionals defined on the boundary. In a second phase a matching parametric grid is imposed. For the ease of presentation the theory is restricted to 2D star-shaped obstacles in low-dimensional setting. We employ the null-field reconstruction technique which is capable of handling large shape deformations. Higher spatial and parametric dimensional cases are discussed, though, not extensively explored in the current study.