Operator expansions and constrained quadratic optimization for interface reconstruction: Impenetrable periodic acoustic media

• A numerical scheme for interface reconstruction of grating shapes in a two-dimensional acoustic medium is proposed.
• The algorithms are stabilized by casting the problem as a constrained quadratic optimization problem.
• Numerical simulations demonstrate the enhanced stability and accuracy of the new approach even in the presence of noise.

Grating scattering is a fundamental model in remote sensing, electromagnetics, ocean acoustics, nondestructive testing, and image reconstruction. In this work, we examine the problem of detecting the geometric properties of gratings in a two-dimensional acoustic medium where the fields are governed by the Helmholtz equation. Building upon our previous Boundary Perturbation approach (implemented with the Operator Expansions formalism) we derive a new approach which augments this with a new “smoothing” mechanism. With numerical simulations we demonstrate the enhanced stability and accuracy of our new approach which further suggests not only a rigorous proof of convergence, but also a path to generalizing the algorithm to multiple layers, three dimensions, and the full equations of linear elasticity and Maxwell’s equations.