The gradient descent method for the convexification to solve boundary value problems of quasi-linear PDEs and a coefficient inverse problem

TL;DR

This paper develops a gradient descent approach with proven global convergence for solving boundary value problems of quasi-linear PDEs and a coefficient inverse problem, using Carleman weights to ensure convexity and handle noisy data.

Contribution

It introduces a new theoretical framework for the global convergence of gradient descent on convex functionals in bounded sets and applies it to nonlinear PDEs and inverse problems.

Findings

Proven Lipschitz convergence rate of the gradient descent method.

Successful numerical solution of a highly nonlinear, ill-posed inverse problem.

Demonstrated robustness to noisy data in boundary value problem solutions.

Abstract

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.