A universal regularization method for ill-posed Cauchy problems for quasilinear partial differential equations

TL;DR

This paper introduces a novel, globally convergent numerical method for solving ill-posed Cauchy problems in quasilinear PDEs, utilizing Carleman Weight Functions to ensure strict convexity of the cost functional.

Contribution

It presents the first globally convergent numerical approach for these problems, leveraging Carleman weights to achieve strict convexity and stability.

Findings

Method demonstrates global convergence in numerical experiments

Carleman weights effectively stabilize the solution process

Applicable to a broad class of quasilinear PDEs

Abstract

For the first time, a globally convergent numerical method is presented for ill-posed Cauchy problems for quasilinear PDEs. The key idea is to use Carleman Weight Functions to construct globally strictly convex Tikhonov-like cost functionals.