Reconstructing the thermal phonon transmission coefficient at solid interfaces in the phonon transport equation

Abstract

The ab initio model for heat propagation is the phonon transport equation, a Boltzmann-like kinetic equation. When two materials are put side by side, the heat that propagates from one material to the other experiences thermal boundary resistance. Mathematically, it is represented by the reflection coefficient of the phonon transport equation on the interface of the two materials. This coefficient takes different values at different phonon frequencies, between different materials. In experiments scientists measure the surface temperature of one material to infer the reflection coefficient as a function of phonon frequency.In this article, we formulate this inverse problem in an optimization framework and apply the stochastic gradient descent (SGD) method for finding the optimal solution. We furthermore prove the maximum principle and show the Lipschitz continuity of the Fréchet derivative. These properties allow us to justify the application of SGD in this setup.

An analysis of stochastic variance reduced gradient for linear inverse problems *

“…One attractive feature of the method is that the computational complexity per iteration does not depend on the data size n, and thus it is directly scalable to large data volume, which is especially attractive in the era of big data. SGD type methods have found applications in several inverse problems, e.g., randomized Kaczmarz method [12,32] in computed tomography, ordered subset expectation maximization [13,21] for positron emission tomography, and more recently also some nonlinear inverse problems, e.g., optical tomography [4] and phonon transmission coefficient recovery [8]. However, the relevant mathematical theory for inverse problems in the lens of regularization theory [7,14,20] is still not fully understood.…”