Student presentation - The reconstruction of space-dependent thermal conductivity from sparse temperature measurements

We present a novel method for reconstructing spatial-dependent thermal conductivity coefficients in 1D and 2D heat equations, leveraging moving sensors that dynamically traverse a domain to record sparse and noisy temperature measurements. Our approach introduces a key innovation: the application of moving sensors significantly enhances reconstruction accuracy by providing temperature data that is far more effective for inferring the thermal conductivity, compared to static measurements. This is derived from the forward problem's sensitivity analysis, which indicates a time interval during which temperature measurements are highly sensitive to reconstruction errors. We then propose the sensors to move in the domain to fully utilize this time interval so that the inverse problems can be solved with fewer sensors and observations while maintaining high reconstruction accuracy. Specifically, we demonstrate the superior performance of our method on the 1D circle and 2D torus, successfully reconstructing thermal conductivity using just one and four moving sensors, respectively, with their trajectories recorded over time. The problem is formulated as a nonlinear least-squares optimization, which minimizes the difference between the predicted temperature measurements based on the reconstructed conductivity and the actual observed measurements, and the gradient of difference is computed efficiently using automatic differentiation, enabling precise updates to the conductivity.