Stability of the reconstruction of the heat reflection coefficient in the phonon transport equation

TL;DR

This paper analyzes the stability of reconstructing the heat reflection coefficient in the phonon transport equation, revealing ill-posedness in the diffusive regime and providing quantitative stability estimates supported by numerical evidence.

Contribution

It provides the first stability analysis of the inverse problem for the reflection coefficient in phonon transport, especially highlighting the ill-posedness as the system approaches the diffusive regime.

Findings

Stability deteriorates as the Knudsen number approaches zero.

The inverse problem becomes ill-posed in the diffusive regime.

Numerical results confirm the theoretical stability estimates.

Abstract

The reflection coefficient is an important thermal property of materials, especially at the nanoscale, and determining this property requires solving an inverse problem based on macroscopic temperature measurements. In this manuscript, we investigate the stability of this inverse problem to infer the reflection coefficient in the phonon transport equation. We show that the problem becomes ill-posed as the system transitions from the ballistic to the diffusive regime, characterized by the Knudsen number converging to zero. Such a stability estimate clarifies the discrepancy observed in previous studies on the well-posedness of this inverse problem. Furthermore, we quantify the rate at which the stability deteriorates with respect to the Knudsen number and confirm the theoretical result with numerical evidence.