An inverse random diffraction grating problem for the Helmholtz equation

Abstract: This paper investigates the inverse scattering problem of time-harmonic plane waves incident on a perfectly reflecting random periodic structure. To simulate random perturbations arising from manufacturing defects and surface wear in real-world grating profiles, we propose a stochastic surface modeling framework motivated by the discretization of the Wiener process. Our approach introduces randomness at discrete nodes and then applies linear interpolation to construct the surface, marking a novel attempt to incorporate the concepts of the Wiener process into random surface representation. Under this framework, each realization of the random surface generates a Lipschitz-continuous diffraction grating, mathematically represented as a sum of a baseline profile and a weighted linear combination of local `tent' basis functions, meanwhile preserving key statistics of the random surface. Building on this representation, we introduce the Recursive Parametric Smoothing Strategy (RPSS) to invert the key statistics of our random surfaces. Combined with Monte Carlo sampling and a wavenumber continuation strategy, our reconstruction scheme demonstrates effectiveness across multiple benchmark scenarios. Several numerical results are presented along with some discussions in the end on reconstruction mechanisms and future extensions.