Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs

TL;DR

This paper introduces a novel numerical method using Carleman weight functions for solving ill-posed Cauchy problems in quasilinear PDEs, proving the existence of solutions and demonstrating numerical effectiveness in 1-D cases.

Contribution

It proves the existence of regularized solutions for a class of ill-posed problems using Carleman weight functions, advancing previous methods that assumed existence.

Findings

Existence of regularized solutions is rigorously proved.

Numerical results validate the method for 1-D quasilinear parabolic PDEs.

Method applies to both Cauchy problems and coefficient inverse problems.

Abstract

In a series of publications of the second author, including some with coauthors, globally strictly convex Tikhonov-like functionals were constructed for some nonlinear ill-posed problems. The main element of such a functional is the presence of the Carleman Weight Function. Compared with previous publications, the main novelty of this paper is that the existence of the regularized solution (i.e. the minimizer) is proved rather than assumed. The method works for both ill-posed Cauchy problems for some quasilinear PDEs of the second order and for some Coefficient Inverse Problems. However, to simplify the presentation, we focus here only on ill-posed Cauchy problems. Along with the theory, numerical results are presented for the case of a 1-D quasilinear parabolic PDE with the lateral Cauchy data given on one edge of the interval (0,1).