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

Guangting Yu, Shiwei Lan, Kookjin Lee, Alex Mahalov

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.

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

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.

Paper Structure

This paper contains 25 sections, 3 theorems, 56 equations, 20 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem background
  3. Recoverability
  4. 1D (special case)
  5. Sensitivity analysis
  6. Extension to 2D space
  7. Sensor Motion, Initial Condition, and External Heat Source
  8. Design of Sensor Motion
  9. Sensor Placement and Movements for D = T
  10. Sensor Placement and Movements for D = T2
  11. Initial Condition and External Heat Source
  12. Overcoming Sparsity in Temperature Measurements
  13. Reduction of Degrees of Freedom (DoF) for D = T
  14. Multiple Sensors for D = T2
  15. Numerical Experiments
  16. ...and 10 more sections

Key Result

Proposition 2.1

Consider the temperature dynamics governed by eqn:heat-dynamics with the initial condition $u_0(\mathbf{x}):=u(t=0, \mathbf{x})$ provided. If the following conditions hold: then the conductivity $a(\mathbf{x})$ cannot be reconstructed from $\{u(t,\mathbf{x})\}_{\mathbb{R^+}\times\mathcal{D}}$, which is the full temperature observation.

Figures (20)

  • Figure 1: Parametrization of $\mathcal{D}$ for the finite difference method
  • Figure 2: Initial positions $s_{1}(0),\cdots,s_{4}(0)$ and trajectories of the four sensors on $\mathbb{T}^2$
  • Figure 3: 1D test case: Heaviside function
  • Figure 4: 1D test case: Piecewise Linear (3-piece, S shape)
  • Figure 5: 1D test case: Piecewise Linear (4-piece, W shape)
  • ...and 15 more figures

Theorems & Definitions (12)

  • Proposition 2.1: Non-recoverability of conductivity
  • proof
  • Remark
  • Lemma 2.2: Lipschitz dependence of solution on conductivity
  • proof
  • Remark
  • Remark
  • Remark
  • Theorem 2.3
  • proof
  • ...and 2 more