Marching schemes for inverse acoustic scattering problems

Since supp f ⊂ Ω, uθ satisfies the plain Helmholtz equation with constant wavenumber outside of Ω, the radiation condition and uθ|∂Ω = gθ. So by solving the exterior problem of the Helmholtz equation (analytically if Ω is a circle), uθ can be computed everywhere outside of Ω without prior knowledge of f . In particular, we can compute h = ∂uθ ∂ν on ∂Ω. Thus, our original problem can be rewritten as: Find an f such that the corresponding Helmholtz equation has a solution uθ with uθ|∂Ω = gθ, ∂uθ ∂ν = hθ on ∂Ω. Let us fix n = 2 for now. Since u is known everywhere outside of Ω, we can safely assume here that Ω is a rectangle (enclosing the original Ω). Then our problem is equivalent (see diagram) to: Define the operator