The Gradient Descent Method for the Convexification to Solve Boundary Value Problems of Quasi-Linear PDEs and a Coefficient Inverse Problem

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