Construction of threshold truncated singular value decomposition algorithm and its application in sound field calculation

One of the main characteristics of ill-posed inverse problems is the strong interference of random noise or measurement errors on the parameters to be estimated. During the solution of inverse problems, measurement errors or noise affect the solution in the form of the reciprocals of singular values, and filtering out the spatial information corresponding to small singular values significantly improves the accuracy of the solution. Due to this reason, noise information in the spatial information corresponding to larger singular values is often ignored. In reality, due to the random distribution of noise, it is distributed across the spaces corresponding to all singular values, not just those corresponding to small singular values. When a relatively large amount of spatial information is retained, noise in the spaces corresponding to larger singular values can significantly interfere with the solution. Therefore, in order to achieve higher computational accuracy, the goal of regularization algorithms should be to filter out noise information from the entire space while retaining spatial information that is highly correlated with the true solution. In response to the aforementioned issues, this paper proposes a Threshold Truncated Singular Value Decomposition (TTSVD) algorithm. Unlike traditional regularization methods, this approach does not filter or truncate spatial vector information based on the magnitude of singular values, but rather truncates it by setting a threshold based on the degree of correlation between spatial vector information and the true solution. This method overcomes the limitations of Tikhonov and truncated singular value decomposition (TSVD) in filtering spatial vector information corresponding to larger singular values, significantly enhancing noise suppression while retaining the main information of the parameters to be estimated. The performance of the TTSVD algorithm is verified through simulation of sound field reconstruction using first-kind Fredholm integral equations, different wave numbers and various signal-to-noise ratios. Finally, the practicality and effectiveness of the algorithm are further validated through sound source localization experiments.