The Reconstruction of the Space-Dependent Thermal Conductivity from Sparse Temperature Measurements

TL;DR

This paper introduces an efficient method for reconstructing space-dependent thermal conductivity in heat equations using moving sensors and automatic differentiation, achieving accurate results with sparse, noisy data in 1D and 2D domains.

Contribution

The paper presents a novel scalable approach combining moving sensors, automatic differentiation, and sampling algorithms for robust conductivity reconstruction from sparse measurements.

Findings

Successful reconstruction on 1D circle and 2D torus

Reduced sensor requirements with high accuracy

Validated robustness against measurement noise

Abstract

We present a novel method for reconstructing the thermal conductivity coefficient in 1D and 2D heat equations using moving sensors that dynamically traverse the domain to record sparse and noisy temperature measurements. We significantly reduce the computational cost associated with forward PDE evaluations by employing automatic differentiation, enabling a more efficient and scalable reconstruction process. This allows the inverse problem to be solved with fewer sensors and observations. Specifically, we demonstrate the successful reconstruction of thermal conductivity on the 1D circle and 2D torus, using one and four moving sensors, respectively, with their positions recorded over time. Our method incorporates sampling algorithms to compute confidence intervals for the reconstructed conductivity, improving robustness against measurement noise. Extensive numerical simulations of heat dynamics validate the efficacy of our approach, confirming both the accuracy and stability of the reconstructed thermal conductivity. Additionally, the method is thoroughly tested using large datasets from machine learning, allowing us to evaluate its performance across various scenarios and ensure its reliability. This approach provides a cost-effective and flexible solution for conductivity reconstruction from sparse measurements, making it a robust tool for solving inverse problems in complex domains.