§ A second degree approximation

On the real line, this predictor-corrector scheme reduces down to Halley's method for finding roots.

For sound soft inverse obstacle scattering the second derivative is computed from

Theorem. Let ∂D be in the class C³. The operator F is two times differentiable on ∂D with second derivative F^"[∂D](h₁,h₂) = u"_∞. The function u"_∞ is the far field pattern of the radiating solution u" in H¹_loc(R²/D) of the exterior Dirichlet problem

Δ u" + k² u" = 0 in R/D

u" =	-h1,ν (u'2)ν - h2,ν (u'1)ν + ( h1,ν h2,ν - h1,th2,t)H uν
	+ ( h1,t(t.∇ h2,t)) + h2,t(t.∇ h1,t)) uν	on ∂D

The two graphs below show gives a comparison of the convergence rate of the two schemes, Newton and Halley; that is the above algorithm using only the predictor (Newton) and then additionally using the corrector (Halley).
Note the very fast convergence rate even of the Newton method and that the first iteration (starting from an initial approximation of a unit circle) of Halley's method itself outperforms the entire Newton scheme.

The two pictures below show reconstructions for the first iteration using both Newton and Halley. The differences between them are thus entirely due to the corrector step. The data was essentially noise free, but an inverse crime was avoided.

Here are the final reconstructions from both methods:

We can also reconstruct under quite large values of data error -- these pictures use Tichonov regularisation in both the predictor and corrector.