An inverse diffraction problem with discrete data (Q1122968)

The author considers the inverse problem which aims at the exploration of the shape of a bounded radiating aperture \(\Omega\) in an infinite plane through some discrete data obtained by measurements on another plane, parallel to the first. The sampling points where the measurements are performed, say \(x_{nm}\) ((n,m)\(\in I)\) are supposed to be known only approximately and lie in some suitable neighbourhoods of points \(\tilde x_{nm}\) of a certain given periodic lattice. Here \(I\subset Z^ 2\) is a finite set of indices. The mathematical statement of the problem is as follows: Given positive numbers \(k^ 2\), \(\epsilon\), \(\delta\), t, real numbers \(\tilde x_{nm}\) and complex numbers \(g_{nm}\), find a bounded set \(\Omega\) such that there exist u(x,t), \(\bar x_{nm}\) and \(\chi_{\Omega}(x)\) satisfying \(\Delta u(x,z)+k^ 2u(x,z)=0\), \(x\in {\mathbb{R}}^ 2\), \(z>0\), \(\sum_{(n,m)\in I}| u(\bar x_{nm},t)- g_{nm}|^ 2\leq \epsilon\), \(| \bar x_{nm}-\tilde x_{nm}| \leq \delta\), (n,m)\(\in I\), \[ (*)\quad \lim_{\rho \to \infty}\int_{S_{\rho}}| \partial u/\partial n-iku|^ 2d\sigma =0,\quad u(x,0)=\chi_{\Omega}(x),\quad (\chi_{\Omega}\equiv 0\quad if\quad x\not\in \Omega). \] Here \(S_{\rho}\) stands for the upper half sphere of radius \(\rho\) while \(\partial /\partial n\) is the derivative in the outer normal direction. It is worthwhile to remark that the statement of the problem, written in the paper, is slightly different from the above-one. For example the function \(\chi_{\Omega}(x)\) is presented by the author as to be the characteristic function of the set \(\Omega\), which necessitates \(\chi_{\Omega}(x)\equiv 1\) for \(x\in \Omega\) (this is not true in general). Similarly, the impinging wave is stated by the author as to be a plane wave travelling from infinity, which is a contradiction, in general, to the radiation condition (*) because u denotes the total field. In fact, the incident field can (and must) be any wave with source lying in the lower half-space \(z<0.\) This problem is a generalization of what was previously considered by the author himself, which corresponds to the case \(\delta =0\) [Numer. Funct. Anal. Optimization 9, 327-357 (1987; Zbl 0632.65124)]. A solution to the problem is obtained through cardinal series. The influence of the number of sampling points as well as the contaminations due to noisy data are discussed.