Laplace approximation in sparse Bayesian learning for structural damage detection

The Bayesian theorem has been demonstrated as a rigorous method for uncertainty assessment and system identification. Given that damage usually occurs at limited positions in the preliminary stage of structural failure, the sparse Bayesian learning has been developed for solving the structural damage detection problem. However, in most cases an analytical posterior probability density function (PDF) of the damage index is not available due to the nonlinear relationship between the measured modal data and structural parameters. This study tackles the nonlinear problem using the Laplace approximation technique. By assuming that the item in the integration follows a Gaussian distribution, the asymptotic solution of the evidence is obtained. Consequently the most probable values of the damage index and hyper-parameters are expressed in a coupled closed form, and then solved sequentially through iterations. The effectiveness of the proposed algorithm is validated using a laboratory tested frame. As compared with other techniques, the present technique results in the analytical solutions of the damage index and hyper-parameters without using hierarchical models or numerical sampling. Consequently, the computation is more efficient.