## 3 Results and Discussion
### The Discussion of the Practical Challenges and Future Advancements  
In practical application scenarios, the AGES-AHK method may encounter challenges such as sensitivity to measurement noise, the tuning of key parameters, and the transition to three-dimensional domains. In real-world applications, noise can introduce errors in time-of-flight measurements, resulting in inaccuracies especially in high-gradient regions where the temperature distribution is more sensitive to small changes in data. In the preliminary work of this study, the effect of noise on reconstruction had been analyzed. The results show that NRMSE grows approximately linearly with increasing random noise. Adding a \(\pm\)1% TOF error increases the average NRMSE by more than 0.4%, which is a sizable increase. This is because the AGES-AHK method is a reallocation of computational resources without altering the underlying reconstruction algorithm LSQR, meaning the method's anti-noise performance is not inherently improved. In future work, implementing noise reduction techniques (e.g., Kalman Filtering) to reduce random noise and regularization techniques (e.g., Tikhonov) to modify the reconstruction algorithm could help reduce the impact of noise on reconstruction accuracy. Furthermore, the AGES-AHK method's performance is highly dependent on the tuning of key parameters, including the hybrid kernel shape parameters and the adaptive grid density in nonuniform discretization. Future work could focus on exploring adaptive methods for parameter tuning. Regarding the dependence on parameter tuning, it is essential to establish a customized parameter library for the target application scenarios through a large number of simulations and experiments. Through this approach, the appropriate parameter base values can be quickly selected within the scope of the application scenario. The optimal parameter values can then be obtained with slight modifications. This parameter library is similar to the common methods in industrial applications, which would not only enhance efficiency but also make the method more accessible for real-world applications. The results in this study demonstrate the effectiveness of the AGES-AHK method in 2D applications. However, extending AGES-AHK to 3D would increase computational costs cubically (O(\(n^{3}\)) grids vs. O(\(n^{2}\))) due to refined meshing and acoustic path calculations. Critical variables affecting performance and accuracy in 3D include the number of transducers, the grid resolution, and the size of the ROI and heat sources. GPU acceleration or parallelized LSQR solvers could mitigate these costs. Additionally, the AGES and AHK methods may face challenges in scenarios with highly nonlinear thermal gradients or irregular temperature distributions. By modifying the grid density distribution to match the temperature field topography, AGES essentially constitutes a trade-off for reallocating computational resources with limited measurement points and grids. In cases where temperature fields exhibit highly irregular or chaotic variations, the reliance on adaptive grid evolution may result in the missing of broader trends. Furthermore, the nonuniform discretization may fail to capture fine details in regions with rapid, non-continuous changes. These limitations highlight the necessity for the integration of higher-order interpolation methods or machine learning techniques to predict optimal grid placements. The finalized adaptive hybrid kernel from the AHK method combines the sharp spatial localization characteristics of G with the global characterization of MQ. However, the Gaussian kernel may fail to fit temperature profiles with drastically excessive gradients or capture the complex interactions among high-gradient regions. The MQ kernel, while showing good global characterization, may fail to capture the broad trends of the reconstruction field in cases where temperature fields exhibit highly irregular or chaotic variations.