Recovering the initial condition in the One-Phase Stefan problem

TL;DR

This paper investigates the problem of reconstructing the initial temperature distribution in a one-dimensional Stefan problem using the melting point position, revealing severe ill-posedness and providing stability estimates and numerical examples.

Contribution

It introduces a new logarithmic stability estimate for the inverse problem, highlighting its severe ill-posedness and employing integral equations and unique continuation properties.

Findings

The inverse problem is severely ill-posed.

A new logarithmic stability estimate is established.

Numerical examples demonstrate the theoretical findings.

Abstract

We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data.