In this paper, a new dimensionality reduction based temperature distribution sensing (TDS) method is proposed to reconstruct the temperature distribution via the limited number of the scattered temperature measurement data. The projective nonnegative matrix factorization (PNMF) method is developed to exact the basis vectors, and the augmented Lagrangian multipliers (ALM) method is proposed to solve the proposed PNMF model. A dimensionality reduction model is obtained via projecting the original temperature distribution onto the spaces spanned by a set of basis. An objective functional that considers the inaccurate properties of the reconstruction model and the measurement data, the Shearlet regularization and the total variation (TV) method is proposed to convert the TDS task into an optimization problem, where the temperature distribution is indirectly reconstructed via solving a low-dimensional vector. An iteration scheme is developed to solve the objective functional. Numerical simulation results validate the feasibility of the proposed reconstruction algorithm.
Conclusion
The acquisition of the temperature distribution from the finite observation data is attractive for real-world applications. Unlike available measurement techniques and inverse heat transfer problems, a dimensionality reduction based method is proposed to achieve the TDS task from the finite number of the temperature measurement data in this paper. The PNMF method is developed to extract the basis vectors from a set of snapshots, and an iterative scheme that integrates the advantages of the ALM method is proposed to solve the proposed PNMF model. A dimensionality reduction model is obtained via projecting the original temperature distribution onto the spaces spanned by a set of basis extracted by the proposed PNMF method. Within the framework of the dimensionality reduction model and the Tikhonov regularization method, a new objective functional that considers the inaccurate properties of the reconstruction model and the measurement data, the Shearlet regularization and the TV method is proposed to convert the TDS task into a minimization problem, where the temperature distribution is indirectly reconstructed by solving a low-dimensional vector using the limited number of the temperature observation data. An iteration scheme that integrates the superiorities of the NMS algorithm is developed to search for the optimal solution of the proposed objective functional. The developed reconstruction method is different from existing local point measurement techniques and tomography-based measurement methods owing to the fact that the temperature distribution can be reconstructed from the finite number of the scattered temperature measurement data. Numerical simulation results validate the feasibility and effectiveness of the proposed algorithm. Furthermore, the reconstruction results from the noise contaminated data show that the proposed algorithm is robust to the inaccuracies of the input data, including the reconstruction model and the measurement data. This study also confirms that the proposed dimensionality reduction model is able to decrease the number of the measurement data for the TDS problem, thereby reducing the number of the sensors and the measurement cost. At a result, a promising method is introduced for the TDS task.
Real-world applications indicate that one reconstruction method may show different numerical performances to various reconstruction tasks, and the determination of an appropriate reconstruction method depends mainly on the measurement requirements, the understandings of the prior knowledge related to the reconstruction objects and the properties of the numerical method. This study provides an alternative approach for the TDS problem, which needs to be further validated by more cases in the future and to be further studied on the respects such as the arrangement and optimization of the measurement points, and the reduction of the inaccuracies of the input data.
The results have been published on Measurement 82 (2016) 176–187.