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