Stability problems in inverse diffraction

Inverse diffraction consists in determining the field distribution on a boundary surface from the knowledge of the distribution on a surface situated within the domain where the wave propagates. This problem is a good example for illustrating the use of least-squares methods (also called regularization methods) for solving linear ill-posed inverse problems. We focus on obtaining error bounds for regularized solutions and show that the stability of the restored field far from the boundary surface is quite satisfactory: the error is proportional to
\varepsilon^{\alpha}(\alpha \simeq 1), \varepsilon
being the error in the data (Hölder continuity). However, the error in the restored field on the boundary surface is only proportional to an inverse power of
\| ln \varepsilon \|
(logarithmic continuity). Such a poor continuity implies some limitations on the resolution which is achievable in practice. In this case, the resolution limit is seen to be about half of the wavelength.