An approximate factorization method for inverse acoustic scattering with phaseless near-field data

This paper is concerned with the inverse acoustic scattering problem with phaseless near-field data at a fixed frequency. An approximate factorization method is developed to numerically reconstruct both the location and shape of the unknown scatterer from the phaseless near-field data generated by incident plane waves at a fixed frequency and measured on the circle $\partial B_R$ with a sufficiently large radius $R$. The theoretical analysis of our method is based on the asymptotic property in the operator norm from $H^{1/2}({\mathbb S}^1)$ to $H^{-1/2}({\mathbb S}^1)$ of the phaseless near-field operator defined in terms of the phaseless near-field data measured on $\partial B_R$ with large enough $R$, where $H^s({\mathbb S}^1)$ is a Sobolev space on the unit circle ${\mathbb S}^1$ for real number $s$, together with the factorization of a modified far-field operator. The asymptotic property of the phaseless near-field operator is also established in this paper with the theory of oscillatory integrals. The unknown scatterer can be either an impenetrable obstacle of sound-soft, sound-hard or impedance type or an inhomogeneous medium with a compact support, and the proposed inversion algorithm does not need to know the boundary condition of the unknown obstacle in advance. Numerical examples are also carried out to demonstrate the effectiveness of our inversion method. To the best of our knowledge, it is the first attempt to develop a factorization type method for inverse scattering problems with phaseless data.