A Perturbation Method for Inverse Scattering in Three-Dimensions Based on the Exact Inverse Scattering Equations

The detection and characterization of macroscopic flaws, such as cracks in solids are fundamental goals of nondestructive evaluation. Many inspection methods use scattered electromagnetic or ultrasonic waves. These methods rely explicitly on the development of inverse scattering theory. This theory seeks to determine the geometrical and material properties of flaws from scattering data

On stability of discretizations of the Helmholtz equation (extended version)

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation