Maximumly weighted iteration for solving inverse problems in dynamics

Abstract

Many problems in dynamics can be formulated as inverse problems that require the determination of the unknown input from the known output. Limited by the measurement noise, the computational cost, the error in numerical computation, etc., solving ill-conditioned inverse problems in dynamics is a big challenge. In this paper, we propose Maximumly Weighted Iteration (MWI) approach to solve ill-conditioned inverse problems in dynamics generally. The ill-condition of the system coefficient matrix is controlled by iterative weighted decomposition and a weighted term, also avoiding matrix inversion of solving and reconstruction process. As an iterative method, MWI has been theoretically proven to have consistency, absolute convergence, and conditional unbiasedness. The numerical results demonstrate that MWI is superior to Truncated Singular Value Decomposition and Tikhonov regularization in terms of accuracy and anti-noise property. Two application cases are used to demonstrate the potential of MWI in engineering practice. The first is elastic vibrations identification combined with randomized resonant metamaterials, and the second is fault diagnosis of mechanical transmission systems.