Research On The Solutions Of Ill-Posed Problems In Sound Identification

The field of inverse problems has existed in many branches of physics and mathematics for a long time. Inverse problems theory has been widely developed within the past decades mainly due to its importance of applications.Inverse problems consist in using the results of actual observations to infer the values of the parameters characterizing the system under investigation. Inverse problems are typically ill-posed in the sense that one of the three items "existence, uniqueness or stability" of the solution may be violated. Therefore some regularization and optimization methods should be involved to suppress the ill-posedness. Actually, a majority of the inverse problems belong to the field of the first-kind Fredholm integral equation, and so does the beamforming mathematic model used for sound identification which this paper studied. Based on Tikhonov regularization method and using the generalized singular value decomposition as the tool for analysis, the main thread of the choice of the regular matrix and the determination of the smoothing parameters is grasped and the in-depth research on the ill-posed problems is carried out. In this paper, we also investigate the cause of ill-posedness, present and analyze the existing numerical algorithms for solving the ill-posed problems, and find that both the matrix L and the regularization parameterλcontrol the smoothness of the regularized solution x_λ. Unfortunately, the existing methods mainly focus on the choice of regularization parameters, and have no idea when the typical choice of matrix L is useless and no useful prior information is available, according to this, we propose a new method for choosing the regular matrix. Tikhonov regularization method has been evolved in this paper.Finally, we apply the algorithm to the actual model, and get an appropriate approximation.