  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 construct an integration grid
whose associated weight function encompasses the irregularities and
nonsmoothness imposed by the random boundary. Thus, the solution can be
evaluated accurately with relatively low number of grid points. The integration
grid is obtained by employing a low-dimensional spatial embedding using the
coarea formula. The proposed method can handle large variation as well as
non-smoothness of the random boundary. For the ease of presentation the theory
is restricted to star-shaped obstacles in low-dimensional setting. Higher
spatial and parametric dimensional cases are discussed, though, not extensively
explored in the current study.
