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

TL;DR

The paper analyzes the stability of reconstructing the boundary reflection coefficient in the Phonon Transport Equation from surface-temperature data. It shows that, under diffusive scaling with Knudsen number $\varepsilon$, the inverse problem becomes exponentially ill-posed as $\varepsilon\to0^+$, with the Lipschitz constant decaying like $e^{-c/\varepsilon}$; this is demonstrated via a singular decomposition separating ballistic and scattering contributions. The main contributions are (1) a precise formulation of the forward PTE model and its non-dimensional diffusive limit, (2) a stability theorem for the inverse problem in terms of $\mathcal{M}^\varepsilon$, and (3) rigorous bounds on the ballistic vs. scattering parts that explain observed discrepancies in prior work, complemented by numerical results showing loss-function flattening in the diffusion limit. The results provide a rigorous explanation for the deterioration of reconstruction stability in nanoscale heat transfer problems and guide experimental design and data interpretation in TDTR-like setups.

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.

Figures (6)

• Figure 1: Schematics of the experimental setup hua2017experimental.
• Figure 2: Cost function $L^\varepsilon$ for different $\varepsilon$ for varying $\eta(\omega)$.
• Figure 3: Two different forms of $\eta(\omega)$ used to compute the approximate values of $\mathcal{M}^\varepsilon(\eta)$.
• Figure 4: Profile of $\Lambda^\varepsilon_{\eta_1}$ (first row) and the difference $\Lambda^\varepsilon_{\eta_1}-\Lambda^\varepsilon_{\eta_2}$ (second row) for varying $\varepsilon$. The $x$-axis indicates the location of the source (in frequency) and the $y$-axis indicates the measurement in time. This plot shows that the sensitivity of the measurement map towards sources centered at various frequencies is severely reduced as $\varepsilon$ decreases. Note the differences changes the order from $O(10^{-7}$ for small $\varepsilon$ to $O(10^{-3})$ for big $\varepsilon$.
• Figure 5: Temperature reading in time, $\Delta T$, for $\eta_1$ and the temperature difference between $\eta_1$ and $\eta_2$, $\Delta T[\eta_1]-\Delta T[\eta_2]$, for $\varepsilon$ being $0.125, 0.5, 4$.

Theorems & Definitions (11)

• Theorem 2
• Corollary 1
• proof
• Theorem 3
• Proposition 4
• Lemma 5: Proposition 3.1 in sun2022unique
• Proposition 6
• Lemma 7
• proof
• proof