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For the first time, we develop in this paper the globally convergentconvexification numerical method for a Coefficient Inverse Problem for the 3DHelmholtz equation for the case when the backscattering data are generated by apoint source running along an interval of a straight line and the wavenumber isfixed. Thus, by varying the wavenumber, one can reconstruct the dielectricconstant depending not only on spatial variables but the wavenumber (i.e.frequency) as well. Our approach relies on a new derivation of a boundary valueproblem for a system of coupled quasilinear elliptic partial differentialequations. This is done via an application of a special truncated Fourier-likemethod. First, we prove the Lipschitz stability estimate for this problem via aCarleman estimate. Next, using the Carleman Weight Function generated by thatestimate, we construct a globally strictly convex cost functional and prove theglobal convergence to the exact solution of the gradient projection method.Finally, our theoretical finding is verified via several numerical tests withcomputationally simulated data. These tests demonstrate that we can accuratelyrecover all three important components of targets of interest: locations,shapes and dielectric constants. In particular, large target/backgroundcontrasts in dielectric constants (up to 10:1) can be accurately calculated.

Convexification for a 3D inverse scattering problem with the moving point source