  This paper is concerned with the study of a version of the globally
convergent convexification method with direct application to synthetic aperture
radar (SAR) imaging. Results of numerical testing are presented for
experimentally collected data for a fake landmine. The SAR imaging technique is
a common tool used to create maps of parts of the surface of the Earth or other
planets. Recently, it has been applied in the context of non-invasive
inspections of buildings in military and civilian services. Nowadays, any SAR
imaging software is based on the Born approximation, which is a linearization
of the original wave-like partial differential equation. One of the essential
assumptions this linearization procedure needs is that only those dielectric
constants are imaged whose values are close to the constant background. In this
work, we propose a radically new idea: to work without any linearization while
still using the same data as the conventional SAR imaging technique uses. We
construct a 2D image of the dielectric constant function using a number of 1D
images of this function obtained via solving a 1D coefficient inverse problem
(CIP) for a hyperbolic equation. Different from our previous studies on the
convexification method with concentration on the global convergence of the
gradient projection method, this time we prove the global convergence of the
gradient descent method, which is easier to implement numerically.
