  We study the global convergence of the gradient descent method of the
minimization of strictly convex functionals on an open and bounded set of a
Hilbert space. Such results are unknown for this type of sets, unlike the case
of the entire Hilbert space. Then, we use our result to establish a general
framework to numerically solve boundary value problems for quasi-linear partial
differential equations (PDEs) with noisy Cauchy data. The procedure involves
the use of Carleman weight functions to convexify a cost functional arising
from the given boundary value problem and thus to ensure the convergence of the
gradient descent method above. We prove the global convergence of the method as
the noise tends to 0. The convergence rate is Lipschitz. Next, we apply this
method to solve a highly nonlinear and severely ill-posed coefficient inverse
problem, which is the so-called back scattering inverse problem. This problem
has many real-world applications. Numerical examples are presented.
