Tikhonov regularized penalty matrix construction method based on the magnitude of singular values and its application in near-field acoustic holography

Tikhonov regularization improves the ill-conditioning of the original transfer matrix by using regularization parameters and a penalty matrix. When the penalty matrix is a unit matrix, Tikhonov regularization is known as ridge estimation. Using the mean value and variance analysis of ridge estimation, it can be observed that the ridge estimation uses a regularization parameter to filter small singular values, and introduces system bias independent of the measurement error. However, due to the large order of magnitude difference between the singular values, the fixed regularization parameter cannot effectively suppress the variance component corresponding to large singular values. As a result, the variance of the estimation increases and the stability of the solution reduces. In view of the above defects, a method is proposed in this paper to construct a regularized penalty matrix based on the magnitude of singular values. Compared with the unit matrix used in ridge estimation, the constructed penalty matrix can introduce an appropriate amount of bias according to the singular values. This reduces the variance of parameter estimation under the condition that the system bias remains approximately unchanged. Finally, the constructed penalty matrix is applied to the source strength solution and sound field reconstruction of equivalent source near-field acoustic holography. The results show that the regularization method using the new penalty matrix enhances the stability of the source strength solution and improves the reconstruction accuracy of the sound field.