A nonlinearly ill-posed problem of reconstructing the temperature from interior data

TL;DR

This paper investigates the challenging problem of reconstructing a temperature distribution within a domain from interior measurements, focusing on a nonlinear elliptic equation that models heat conduction.

Contribution

It introduces a novel approach to address the nonlinear ill-posed inverse problem of temperature reconstruction from interior data.

Findings

Demonstrates the stability of the reconstruction method under certain conditions

Provides theoretical analysis of the ill-posedness of the nonlinear problem

Proposes a regularization technique for practical reconstruction

Abstract

We consider the problem of reconstructing, from the interior data $u(x,1)$, a function $u$ satisfying a nonlinear elliptic equation $$ \Delta u = f(x,y,u(x,y)), x \in \RR, y > 0. $$