Convexification for a Three-Dimensional Inverse Scattering Problem with the Moving Point Source

For the first time, we develop in this paper the globally convergent convexification numerical method for a coefficient inverse problem for the three-dimensional 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 on the wavenumber (i.e., frequency) as well. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear 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.