We consider an inverse source problem for the Helmholtz equation. This is concerned with the reconstruction of an unknown source from multi-frequency data obtained from the radiated fields. Based on a Fourier expansion of the source, a numerical method is proposed to solve the inverse problem. Stability is analyzed and numerical experiments are presented to show the effectiveness of our method.

We consider an inverse source problem of determining a source term in the Helmholtz equation from multi-frequency far-field measurements. Based on the Fourier series expansion, we develop a novel non-iterative reconstruction method for solving the problem. A promising feature of this method is that it utilizes the data from only a few observation directions for each frequency. Theoretical uniqueness and stability analysis are provided. Numerical experiments are conducted to illustrate the effectiveness and efficiency of the proposed method in both two and three dimensions.

This paper is devoted to the uniqueness in inverse acoustic scattering problems for the Helmholtz equation with phaseless far-field data. Some novel techniques are developed to overcome the difficulty of translation invariance induced by a single incident plane wave. In this paper, based on adding a reference ball as an extra artificial impenetrable obstacle (resp. penetrable homogeneous medium) to the inverse obstacle (resp. medium) scattering system and then using superpositions of a fixed plane wave and some point sources as the incident waves, we rigorously prove that the location and shape of the obstacle as well as its boundary condition or the refractive index can be uniquely determined by the modulus of far-field patterns. The reference ball technique in conjunction with the superposition of incident waves brings in several salient benefits. First, the framework of our theoretical analysis can be applied to both the inverse obstacle and medium scattering problems. Second, for inverse obstacle scattering, the underlying boundary condition could be of a general type and be uniquely determined. Finally, only a single frequency is needed.

This paper is concerned with the inverse source problem of reconstructing an unknown acoustic excitation from phaseless measurements of the radiated fields away at multiple frequencies. It is well known that the non-uniqueness issue is a major challenge associated with such an inverse problem. We develop a novel strategy to overcome this challenging problem by recovering the radiated fields via adding some reference point sources as extra artificial sources to the inverse source system. This novel reference source technique requires only a few extra data, and brings in a simple phase retrieval formula. The stability of this phase retrieval approach is rigorously analyzed. After the reacquisition of the phase information, the multi-frequency inverse source problem with recovered phase information is solved by the Fourier method, which is non-iterative, fast and easy to implement. Several numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed method.

In this paper, we consider the inverse problem of determining the location and the shape of a sound-soft obstacle from the modulus of the far-field data for a single incident plane wave. By adding a reference ball artificially to the inverse scattering system, we propose a system of nonlinear integral equations based iterative scheme to reconstruct both the location and the shape of the obstacle. The reference ball technique causes few extra computational costs, but breaks the translation invariance and brings information about the location of the obstacle. Several validating numerical examples are provided to illustrate the effectiveness and robustness of the proposed inversion algorithm.