Abstract: The well known translation invariance property of the phaseless far field patterns with incident plane waves make it is impossible to reconstruct the location of the underlying scatterers even for multiple incident directions and frequencies. To overcome this difficulty, a fast and stable phase retrieval approach is proposed based on a simple geometrical result which provides a stable reconstruction of a point in the plane from three distances to the given points. Both the uniqueness and direct sampling methods will also be mentioned in this talk. This talk is based on the following joint works:

1. X. Ji and X. Liu, Inverse elastic scattering problems with phaseless far field data, in Special Issue in Memory of Professor Armin Lechleiter, 1982-2018, Inverse Problems , 2019.

2. X. Ji, Y. Jia and X. Liu, Inverse fluid-solid interaction scattering problem using phased and phaseless far field data, AMAS-Special Issue on Inverse Problems, 2019.

3. X. Ji, X. Liu and B. Zhang, Inverse acoustic scattering with phaseless far field data: uniqueness, phase retrieval and direct sampling methods, SIAM J. Imaging Sci., 12(2) 2019, 1163-1189.

4. X. Ji, X. Liu and B. Zhang, Phaseless inverse source scattering problem: phase retrieval, uniqueness and direct sampling methods, J. Comput. Phys.: X1, 2019, 100003.

5. X. Ji, X. Liu and B. Zhang, Target reconstruction with a reference point scatterer using phaseless far field patterns, SIAM J. Imaging Sci., 12(1), 2019, 372-391.