The Carleman-Newton method to globally reconstruct a source term for nonlinear parabolic equation

TL;DR

This paper introduces the Carleman-Newton method, a novel algorithm combining Carleman estimates and Newton's method, to reliably reconstruct source terms in nonlinear parabolic equations from boundary data, overcoming stability issues.

Contribution

The paper develops the Carleman-Newton method, a new approach that does not require a good initial guess and is computationally efficient for solving inverse source problems in nonlinear parabolic PDEs.

Findings

The method achieves stable reconstruction from boundary data.

Numerical examples demonstrate effectiveness and efficiency.

The approach improves reliability over traditional least squares methods.

Abstract

We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of Fourier coefficients of the solution to the parabolic governing equation. We solve this system by the Carleman-Newton method. The Carleman-Newton method is a newly developed algorithm to solve nonlinear PDEs. The strength of the Carleman-Newton method includes (1) no good initial guess is required and (2) the computational cost is not expensive. These features are rigorously proved. Having the solutions to this system in hand, we can directly compute the solution to the proposed inverse problem. Some numerical examples are displayed.