Comparative studies of Tikhonov regularization and truncated singular value decomposition in the three-dimensional flame temperature field reconstruction

Abstract

Radiative imaging of combustion flame in furnace of power plant plays an increasingly important role in combustion diagnosis. The flame radiation image taken by a charge-coupled device (CCD) camera can reconstruct three-dimensional flame temperature distribution in the furnace. CCD cameras are used for capturing the flame images to obtain the line-of-sight radiation intensities. The temperature reconstruction matrix equation is a seriously pathological equation. Thus the temperature field reconstruction problem is an ill-posed problem. The two algorithms (Tikhonov regularization and truncated singular value decomposition (TSVD)) for solving the temperature field reconstruction are introduced. The size of the numerical simulation system is 10 m × 10 m × 10 m, which is divided into 10 × 10 × 10 volume elements in the three dimensions. Each volume element is a unit cube. Generalized cross-validation (GCV) is used to select the correct regularization parameter. The measured data are simulated by adding different random errors to the exact solution of the direct problem. The reconstructed temperature deviation is calculated by the two algorithms separately. When the measuring errors are 0.05 and 0.10, the reconstruction errors based on Tikhonov are respectively 19.3% and 7.0%, less than those based on TSVD. When the measuring errors are 0, 0.01, 0.03 and 0.07, the differences between the two kinds of errors are all less than 3%. Both the algorithms can reconstruct the correct temperature field. The times required to reconstruct the temperature field by the two algorithms are compared and their effects of the maximum temperature are also compared. When the measuring errors are 0, 0.01, 0.03, 0.05, 0.07 and 0.1, the reconstruction times based on Tikhonov are respectively-0.0917,-0.049, 0.161, 0.002, 0.135 and 0.091 s, shorter than the reconstruction times based on TSVD. There is singular value decomposition (SVD) in TSVD. And this process takes more than 2 s. If the problem is more complicated, SVD takes much more time. The errors of the maximum reconstruction temperature under Tikhonov are smaller. And the position of the maximum reconstruction temperature under Tikhonov is near the position of the exact maximum temperature in space. The maximum reconstruction temperature under TSVD is not so good as that under Tikhonov. Preliminary results indicate that the Tikhonov-based reconstruction is slightly better than the TSVD-based reconstruction, especially in reconstruction error, reconstruction time, and effects of the maximum temperature.